Weight distribution of random linear codes and Krawtchouk polynomials

Random Structures and Algorithms Pub Date : 2024-03-15 DOI:10.1002/rsa.21214

Alex Samorodnitsky

{"title":"Weight distribution of random linear codes and Krawtchouk polynomials","authors":"Alex Samorodnitsky","doi":"10.1002/rsa.21214","DOIUrl":null,"url":null,"abstract":"For <mjx-container aria-label=\"0 less than lamda less than 1\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,2,4\" data-semantic-content=\"1,3\" data-semantic- data-semantic-role=\"inequality\" data-semantic-speech=\"0 less than lamda less than 1\" data-semantic-type=\"relseq\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"relseq,<\" data-semantic-parent=\"5\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,<\" data-semantic-parent=\"5\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/0b6af63e-f89a-49ec-8e43-239794a37806/rsa21214-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,2,4\" data-semantic-content=\"1,3\" data-semantic-role=\"inequality\" data-semantic-speech=\"0 less than lamda less than 1\" data-semantic-type=\"relseq\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn><mo data-semantic-=\"\" data-semantic-operator=\"relseq,<\" data-semantic-parent=\"5\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\"><</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"5\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\">λ</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,<\" data-semantic-parent=\"5\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\"><</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn></mrow>$$ 0<\\lambda <1 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> and <mjx-container aria-label=\"n right arrow infinity\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"arrow\" data-semantic-speech=\"n right arrow infinity\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,→\" data-semantic-parent=\"3\" data-semantic-role=\"arrow\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"unknown\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/d8e72604-98a2-45c0-a40b-fa336b10c07b/rsa21214-math-0002.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic-role=\"arrow\" data-semantic-speech=\"n right arrow infinity\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,→\" data-semantic-parent=\"3\" data-semantic-role=\"arrow\" data-semantic-type=\"relation\">→</mo><mi data-semantic-=\"\" data-semantic-parent=\"3\" data-semantic-role=\"unknown\" data-semantic-type=\"identifier\">∞</mi></mrow>$$ n\\to \\infty $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> pick uniformly at random <mjx-container aria-label=\"lamda n\" ctxtmenu_counter=\"2\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"0,1\" data-semantic-content=\"2\" data-semantic- data-semantic-role=\"implicit\" data-semantic-speech=\"lamda n\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"3\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/b03f26e0-ef1d-4b70-9ec6-13fbdc00d0d8/rsa21214-math-0003.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"0,1\" data-semantic-content=\"2\" data-semantic-role=\"implicit\" data-semantic-speech=\"lamda n\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\">λ</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"3\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\">⁢</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow>$$ \\lambda n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> vectors in <mjx-container aria-label=\"StartSet 0 comma 1 EndSet Superscript n Baseline\" ctxtmenu_counter=\"3\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"9\" data-semantic-content=\"7\" data-semantic- data-semantic-role=\"unknown\" data-semantic-speech=\"StartSet 0 comma 1 EndSet Superscript n Baseline\" data-semantic-type=\"postfixop\"><mjx-mrow data-semantic-children=\"8\" data-semantic-content=\"0,4\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"set collection\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"9\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"1,2,3\" data-semantic-content=\"2\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"punctuated\" data-semantic-parent=\"8\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\" rspace=\"3\" style=\"margin-left: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"9\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow><mjx-msup data-semantic-children=\"5,6\" data-semantic-embellished=\"operator\" data-semantic- data-semantic-operator=\"postfixop, \" data-semantic-parent=\"10\" data-semantic-role=\"unknown\" data-semantic-type=\"superscript\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-mo data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"unknown\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-script style=\"vertical-align: 0.363em;\"><mjx-mrow size=\"s\"><mjx-mi data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/fac885e3-0e4e-48bd-aad7-6d5564a7538e/rsa21214-math-0004.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"9\" data-semantic-content=\"7\" data-semantic-role=\"unknown\" data-semantic-speech=\"StartSet 0 comma 1 EndSet Superscript n Baseline\" data-semantic-type=\"postfixop\"><mrow data-semantic-=\"\" data-semantic-children=\"8\" data-semantic-content=\"0,4\" data-semantic-parent=\"10\" data-semantic-role=\"set collection\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"9\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">{</mo><mrow data-semantic-=\"\" data-semantic-children=\"1,2,3\" data-semantic-content=\"2\" data-semantic-parent=\"9\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"8\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn><mo data-semantic-=\"\" data-semantic-operator=\"punctuated\" data-semantic-parent=\"8\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\">,</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"8\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn></mrow><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"9\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">}</mo></mrow><msup data-semantic-=\"\" data-semantic-children=\"5,6\" data-semantic-embellished=\"operator\" data-semantic-operator=\"postfixop, \" data-semantic-parent=\"10\" data-semantic-role=\"unknown\" data-semantic-type=\"superscript\"><mo data-semantic-=\"\" data-semantic-parent=\"7\" data-semantic-role=\"unknown\" data-semantic-type=\"operator\"> </mo><mrow><mi data-semantic-=\"\" data-semantic-font=\"italic\" data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow></msup></mrow>$$ {\\left\\{0,1\\right\\}}^n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> and let <mjx-container aria-label=\"upper C\" ctxtmenu_counter=\"4\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper C\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/660e324d-9650-4232-a609-8b50ef6b600c/rsa21214-math-0005.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper C\" data-semantic-type=\"identifier\">C</mi></mrow>$$ C $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> be the orthogonal complement of their span. Given <mjx-container aria-label=\"0 less than gamma less than one half\" ctxtmenu_counter=\"5\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,2,6\" data-semantic-content=\"1,3\" data-semantic- data-semantic-role=\"inequality\" data-semantic-speech=\"0 less than gamma less than one half\" data-semantic-type=\"relseq\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"relseq,<\" data-semantic-parent=\"7\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,<\" data-semantic-parent=\"7\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mfrac data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"4,5\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"vulgar\" data-semantic-type=\"fraction\"><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/d9552daf-b011-4a26-b338-3138a088e4bc/rsa21214-math-0006.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,2,6\" data-semantic-content=\"1,3\" data-semantic-role=\"inequality\" data-semantic-speech=\"0 less than gamma less than one half\" data-semantic-type=\"relseq\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn><mo data-semantic-=\"\" data-semantic-operator=\"relseq,<\" data-semantic-parent=\"7\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\"><</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"7\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\">γ</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,<\" data-semantic-parent=\"7\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\"><</mo><mfrac data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"4,5\" data-semantic-parent=\"7\" data-semantic-role=\"vulgar\" data-semantic-type=\"fraction\"><mrow><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn></mrow><mrow><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn></mrow></mfrac></mrow>$$ 0<\\gamma <\\frac{1}{2} $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> with <mjx-container aria-label=\"0 less than lamda less than h left parenthesis gamma right parenthesis\" ctxtmenu_counter=\"6\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,2,10\" data-semantic-content=\"1,3\" data-semantic- data-semantic-role=\"inequality\" data-semantic-speech=\"0 less than lamda less than h left parenthesis gamma right parenthesis\" data-semantic-type=\"relseq\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"11\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"relseq,<\" data-semantic-parent=\"11\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"11\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,<\" data-semantic-parent=\"11\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"4,8\" data-semantic-content=\"9,4\" data-semantic- data-semantic-parent=\"11\" data-semantic-role=\"simple function\" data-semantic-type=\"appl\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"6\" data-semantic-content=\"5,7\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"8\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"8\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/d77d76a2-1ba5-4c47-86d4-2d01e9dad60c/rsa21214-math-0007.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,2,10\" data-semantic-content=\"1,3\" data-semantic-role=\"inequality\" data-semantic-speech=\"0 less than lamda less than h left parenthesis gamma right parenthesis\" data-semantic-type=\"relseq\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"11\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn><mo data-semantic-=\"\" data-semantic-operator=\"relseq,<\" data-semantic-parent=\"11\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\"><</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"11\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\">λ</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,<\" data-semantic-parent=\"11\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\"><</mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"4,8\" data-semantic-content=\"9,4\" data-semantic-parent=\"11\" data-semantic-role=\"simple function\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">h</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\">⁡</mo><mrow data-semantic-=\"\" data-semantic-children=\"6\" data-semantic-content=\"5,7\" data-semantic-parent=\"10\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"8\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"8\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\">γ</mi><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"8\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow></mrow>$$ 0<\\lambda <h\\left(\\gamma \\right) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, let <mjx-container aria-label=\"upper X\" ctxtmenu_counter=\"7\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/b32904a8-e72c-4ad3-ad1a-de6b3d5e6138/rsa21214-math-0008.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\">X</mi></mrow>$$ X $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> be the random variable that counts the number of words in <mjx-container aria-label=\"upper C\" ctxtmenu_counter=\"8\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper C\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/940c037d-91e0-410c-a143-f706a075d1a9/rsa21214-math-0009.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper C\" data-semantic-type=\"identifier\">C</mi></mrow>$$ C $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> of Hamming weight <mjx-container aria-label=\"i equals gamma n\" ctxtmenu_counter=\"9\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,5\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"equality\" data-semantic-speech=\"i equals gamma n\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,=\" data-semantic-parent=\"6\" data-semantic-role=\"equality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"2,3\" data-semantic-content=\"4\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"5\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/cefefa00-b758-4d8d-9262-cf9de6b79eee/rsa21214-math-0010.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,5\" data-semantic-content=\"1\" data-semantic-role=\"equality\" data-semantic-speech=\"i equals gamma n\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">i</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,=\" data-semantic-parent=\"6\" data-semantic-role=\"equality\" data-semantic-type=\"relation\">=</mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"2,3\" data-semantic-content=\"4\" data-semantic-parent=\"6\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"5\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\">γ</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"5\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\">⁢</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow></mrow>$$ i=\\gamma n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> (where <mjx-container aria-label=\"i\" ctxtmenu_counter=\"10\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"i\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/c159f13e-eefe-4c85-aa03-ce65262bc1b3/rsa21214-math-0011.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"i\" data-semantic-type=\"identifier\">i</mi></mrow>$$ i $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> is assumed to be an even integer). Linial and Mosheiff [Random Struct. Algorithms. 62 (2023), 68Ű-130] determined the asymptotics of the moments of <mjx-container aria-label=\"upper X\" ctxtmenu_counter=\"11\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/908b84cf-f5c8-42d0-9d2b-9b6016f8d934/rsa21214-math-0012.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\">X</mi></mrow>$$ X $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> of all orders <mjx-container aria-label=\"o left parenthesis StartFraction n Over log n EndFraction right parenthesis\" ctxtmenu_counter=\"12\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,9\" data-semantic-content=\"10,0\" data-semantic- data-semantic-role=\"simple function\" data-semantic-speech=\"o left parenthesis StartFraction n Over log n EndFraction right parenthesis\" data-semantic-type=\"appl\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"11\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"11\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"6\" data-semantic-content=\"7,8\" data-semantic- data-semantic-parent=\"11\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"9\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow><mjx-mfrac data-semantic-children=\"1,5\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"division\" data-semantic-type=\"fraction\"><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,3\" data-semantic-content=\"4,2\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"prefix function\" data-semantic-type=\"appl\" size=\"s\"><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"5\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"5\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mrow><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"9\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/babe71d2-7ed4-4bd2-8023-48a36fa9efbf/rsa21214-math-0013.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,9\" data-semantic-content=\"10,0\" data-semantic-role=\"simple function\" data-semantic-speech=\"o left parenthesis StartFraction n Over log n EndFraction right parenthesis\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"11\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">o</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"11\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\">⁡</mo><mrow data-semantic-=\"\" data-semantic-children=\"6\" data-semantic-content=\"7,8\" data-semantic-parent=\"11\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"fenced\" data-semantic-parent=\"9\" data-semantic-role=\"open\" data-semantic-type=\"fence\">(</mo><mrow><mfrac data-semantic-=\"\" data-semantic-children=\"1,5\" data-semantic-parent=\"9\" data-semantic-role=\"division\" data-semantic-type=\"fraction\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,3\" data-semantic-content=\"4,2\" data-semantic-parent=\"6\" data-semantic-role=\"prefix function\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-font=\"normal\" data-semantic-operator=\"appl\" data-semantic-parent=\"5\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\">log</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"5\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\">⁡</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow></mfrac></mrow><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"fenced\" data-semantic-parent=\"9\" data-semantic-role=\"close\" data-semantic-type=\"fence\">)</mo></mrow></mrow>$$ o\\left(\\frac{n}{\\log n}\\right) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. In this paper we extend their estimates up to moments of linear order. Our key observation is that the behavior of the suitably normalized <mjx-container aria-label=\"k Superscript t h\" ctxtmenu_counter=\"13\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-msup data-semantic-children=\"0,4\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"k Superscript t h\" data-semantic-type=\"superscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: 0.363em;\"><mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"1,2\" data-semantic-content=\"3\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\" size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"4\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/c5e4d402-983c-40b6-9d01-34d0c43f2385/rsa21214-math-0014.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup data-semantic-=\"\" data-semantic-children=\"0,4\" data-semantic-role=\"latinletter\" data-semantic-speech=\"k Superscript t h\" data-semantic-type=\"superscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">k</mi></mrow><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"1,2\" data-semantic-content=\"3\" data-semantic-parent=\"5\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">t</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"4\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\">⁢</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">h</mi></mrow></msup></mrow>$$ {k}^{th} $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> moment of <mjx-container aria-label=\"upper X\" ctxtmenu_counter=\"14\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/302eec25-3e17-41a1-bda5-3b644430793d/rsa21214-math-0015.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\">X</mi></mrow>$$ X $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> is essentially determined by the <mjx-container aria-label=\"k Superscript t h\" ctxtmenu_counter=\"15\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-msup data-semantic-children=\"0,4\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"k Superscript t h\" data-semantic-type=\"superscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: 0.363em;\"><mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"1,2\" data-semantic-content=\"3\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\" size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"4\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/c24279fb-bd1a-4e25-920d-0904435d9f15/rsa21214-math-0016.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup data-semantic-=\"\" data-semantic-children=\"0,4\" data-semantic-role=\"latinletter\" data-semantic-speech=\"k Superscript t h\" data-semantic-type=\"superscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">k</mi></mrow><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"1,2\" data-semantic-content=\"3\" data-semantic-parent=\"5\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">t</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"4\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\">⁢</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">h</mi></mrow></msup></mrow>$$ {k}^{th} $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> norm of the Krawtchouk polynomial <mjx-container aria-label=\"upper K Subscript i\" ctxtmenu_counter=\"16\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper K Subscript i\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.04em;\"><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/5590113d-41a9-47a6-bf00-6dcc05e9c710/rsa21214-math-0017.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper K Subscript i\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">K</mi></mrow><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">i</mi></mrow></msub></mrow>$$ {K}_i $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>.","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"367 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Random Structures and Algorithms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1002/rsa.21214","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

Abstract

For

0 < λ < 1 $$ 0<\lambda <1 $$

and

n \to \infty $$ n\to \infty $$

pick uniformly at random

λ n $$ \lambda n $$

vectors in

{0, 1}^{n} $$ {\left\{0,1\right\}}^n $$

and let

C $$ C $$

be the orthogonal complement of their span. Given

0 < γ < \frac{1}{2} $$ 0<\gamma <\frac{1}{2} $$

with

0 < λ < h (γ) $$ 0<\lambda <h\left(\gamma \right) $$

, let

X $$ X $$

be the random variable that counts the number of words in

C $$ C $$

of Hamming weight

i = γ n $$ i=\gamma n $$

(where

i $$ i $$

is assumed to be an even integer). Linial and Mosheiff [Random Struct. Algorithms. 62 (2023), 68Ű-130] determined the asymptotics of the moments of

X $$ X $$

of all orders

o (\frac{n}{\log n}) $$ o\left(\frac{n}{\log n}\right) $$

. In this paper we extend their estimates up to moments of linear order. Our key observation is that the behavior of the suitably normalized

k^{t h} $$ {k}^{th} $$

moment of

X $$ X $$

is essentially determined by the

k^{t h} $$ {k}^{th} $$

norm of the Krawtchouk polynomial

K_{i} $$ {K}_i $$

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随机线性编码的权重分布和 Krawtchouk 多项式

对于 0<λ<1$$ 0<\lambda <1 $$ 和 n→∞$$ n\to \infty $$ 在{0,1} n$$ {\left\{0,1\right}}^n$ 中均匀随机地选取 λn$$ \lambda n $$ 矢量，并让 C$$ C $$ 是它们跨度的正交补集。给定 0<γ<12$$ 0<\gamma <\frac{1}{2} $$ 与 0<λ<h(γ)$$ 0<\lambda <；h\left(\gamma \right) $$，设 X$$ X $$为随机变量，用于计算 C$$ C $ 中汉明权重为 i=γn$$ i=gamma n $$$（其中 i$$ i $$假定为偶整数）的单词数。Linial 和 Mosheiff [Random Struct. Algorithms.在本文中，我们将其估计值扩展到线性阶矩。我们的主要观察结果是，X$$ X $$ 的适当归一化第 k$$ {k}^{th} $$ 矩的行为基本上由 Krawtchouk 多项式 Ki$$ {K}_i $$ 的第 k$$ {k}^{th} $$ 规范决定。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Random Structures and Algorithms

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