Pub Date : 2020-10-18DOI: 10.26493/1855-3974.2193.0b0
Mark S. MacLean, Štefko Miklavič
Let Γ denote a non-bipartite distance-regular graph with vertex set X , diameter D ≥ 3 , and valency k ≥ 3 . Fix x ∈ X and let T = T ( x ) denote the Terwilliger algebra of Γ with respect to x . For any z ∈ X and for 0 ≤ i ≤ D , let Γ i ( z ) = { w ∈ X : ∂( z , w ) = i }. For y ∈ Γ 1 ( x ) , abbreviate D j i = D j i ( x , y ) = Γ i ( x ) ∩ Γ j ( y ) (0 ≤ i , j ≤ D ) . For 1 ≤ i ≤ D and for a given y , we define maps H i : D i i → ℤ and V i : D i − 1 i ∪ D i i − 1 → ℤ as follows: H i ( z ) = | Γ 1 ( z ) ∩ D i − 1 i − 1 |, V i ( z ) = | Γ 1 ( z ) ∩ D i − 1 i − 1 |. We assume that for every y ∈ Γ 1 ( x ) and for 2 ≤ i ≤ D , the corresponding maps H i and V i are constant, and that these constants do not depend on the choice of y . We further assume that the constant value of H i is nonzero for 2 ≤ i ≤ D . We show that every irreducible T -module of endpoint 1 is thin. Furthermore, we show Γ has exactly three irreducible T -modules of endpoint 1, up to isomorphism, if and only if three certain combinatorial conditions hold. As examples, we show that the Johnson graphs J ( n , m ) where n ≥ 7, 3 ≤ m < n /2 satisfy all of these conditions.
设Γ表示顶点集X，直径D≥3，价k≥3的非二部距离正则图。固定x∈x，设T = T (x)表示Γ关于x的Terwilliger代数。对于任何z∈X 0≤我≤D,让Γ我(z) = {w∈X:∂z, w =我}。y∈Γ1 (x),缩写D j D i =我(x, y) =Γ(x)∩Γj (y)(0≤i, j≤D)。1≤≤D和对于一个给定的y,我们定义地图H i: D我→ℤ和V我:我∪−1 D我−1→ℤ如下:H (z) = |Γ1 (z)∩D我−1−1 |,V (z) = |Γ1 (z)∩D我−1−1 |。我们假设对于每一个y∈Γ 1 (x)，对于2≤i≤D，对应的映射H i和V i是常数，并且这些常数不依赖于y的选择。进一步假设当2≤i≤D时，H i的常数不为零。我们证明了端点1的每个不可约T模都是薄模。更进一步，我们证明Γ有三个端点1的不可约T模，直到同构，当且仅当三个特定的组合条件成立。作为例子，我们证明了其中n≥7,3≤m < n /2的Johnson图J (n, m)满足所有这些条件。
{"title":"On a certain class of 1-thin distance-regular graphs","authors":"Mark S. MacLean, Štefko Miklavič","doi":"10.26493/1855-3974.2193.0b0","DOIUrl":"https://doi.org/10.26493/1855-3974.2193.0b0","url":null,"abstract":"Let Γ denote a non-bipartite distance-regular graph with vertex set X , diameter D ≥ 3 , and valency k ≥ 3 . Fix x ∈ X and let T = T ( x ) denote the Terwilliger algebra of Γ with respect to x . For any z ∈ X and for 0 ≤ i ≤ D , let Γ i ( z ) = { w ∈ X : ∂( z , w ) = i }. For y ∈ Γ 1 ( x ) , abbreviate D j i = D j i ( x , y ) = Γ i ( x ) ∩ Γ j ( y ) (0 ≤ i , j ≤ D ) . For 1 ≤ i ≤ D and for a given y , we define maps H i : D i i → ℤ and V i : D i − 1 i ∪ D i i − 1 → ℤ as follows: H i ( z ) = | Γ 1 ( z ) ∩ D i − 1 i − 1 |, V i ( z ) = | Γ 1 ( z ) ∩ D i − 1 i − 1 |. We assume that for every y ∈ Γ 1 ( x ) and for 2 ≤ i ≤ D , the corresponding maps H i and V i are constant, and that these constants do not depend on the choice of y . We further assume that the constant value of H i is nonzero for 2 ≤ i ≤ D . We show that every irreducible T -module of endpoint 1 is thin. Furthermore, we show Γ has exactly three irreducible T -modules of endpoint 1, up to isomorphism, if and only if three certain combinatorial conditions hold. As examples, we show that the Johnson graphs J ( n , m ) where n ≥ 7, 3 ≤ m < n /2 satisfy all of these conditions.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90241225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-15DOI: 10.26493/1855-3974.1907.3C2
T. Komatsu
Carlitz defined the degenerate Bernoulli polynomials β n ( λ , x ) by means of the generating function t ((1 + λ t ) 1/ λ − 1) −1 (1 + λ t ) x / λ . In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. In this paper, we show some expressions and properties of hypergeometric degenerate Bernoulli polynomials β N , n ( λ , x ) and numbers, in particular, in terms of determinants. The coefficients of the polynomial β n ( λ , 0) were completely determined by Howard in 1996. We determine the coefficients of the polynomial β N , n ( λ , 0) . Hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers appear in the coefficients.
Carlitz用生成函数t ((1 + λ t) 1/ λ−1)- 1 (1 + λ t) x / λ定义了退化伯努利多项式β n (λ， x)。1875年，格莱舍给出了数的几个有趣的行列式，包括伯努利数、柯西数和欧拉数。本文给出了超几何简并伯努利多项式β N, N (λ， x)和数的一些表达式和性质，特别是关于行列式的性质。1996年Howard完全确定了多项式β n (λ， 0)的系数。我们确定了多项式β N, N (λ， 0)的系数。系数中出现了超几何伯努利数和超几何柯西数。
{"title":"Hypergeometric degenerate Bernoulli polynomials and numbers","authors":"T. Komatsu","doi":"10.26493/1855-3974.1907.3C2","DOIUrl":"https://doi.org/10.26493/1855-3974.1907.3C2","url":null,"abstract":"Carlitz defined the degenerate Bernoulli polynomials β n ( λ , x ) by means of the generating function t ((1 + λ t ) 1/ λ − 1) −1 (1 + λ t ) x / λ . In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. In this paper, we show some expressions and properties of hypergeometric degenerate Bernoulli polynomials β N , n ( λ , x ) and numbers, in particular, in terms of determinants. The coefficients of the polynomial β n ( λ , 0) were completely determined by Howard in 1996. We determine the coefficients of the polynomial β N , n ( λ , 0) . Hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers appear in the coefficients.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73730987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-05DOI: 10.26493/1855-3974.2465.571
Selim Bahadır, T. Ekim, Didem Gözüpek
A subset of vertices in a graph is called a total dominating set if every vertex of the graph is adjacent to at least one vertex of this set. A total dominating set is called minimal if it does not properly contain another total dominating set. In this paper, we study graphs whose all minimal total dominating sets have the same size, referred to as well-totally-dominated (WTD) graphs. We first show that WTD graphs with bounded total domination number can be recognized in polynomial time. Then we focus on WTD graphs with total domination number two. In this case, we characterize triangle-free WTD graphs and WTD graphs with packing number two, and we show that there are only finitely many planar WTD graphs with minimum degree at least three. Lastly, we show that if the minimum degree is at least three then the girth of a WTD graph is at most 12. We conclude with several open questions.
{"title":"Well-totally-dominated graphs","authors":"Selim Bahadır, T. Ekim, Didem Gözüpek","doi":"10.26493/1855-3974.2465.571","DOIUrl":"https://doi.org/10.26493/1855-3974.2465.571","url":null,"abstract":"A subset of vertices in a graph is called a total dominating set if every vertex of the graph is adjacent to at least one vertex of this set. A total dominating set is called minimal if it does not properly contain another total dominating set. In this paper, we study graphs whose all minimal total dominating sets have the same size, referred to as well-totally-dominated (WTD) graphs. We first show that WTD graphs with bounded total domination number can be recognized in polynomial time. Then we focus on WTD graphs with total domination number two. In this case, we characterize triangle-free WTD graphs and WTD graphs with packing number two, and we show that there are only finitely many planar WTD graphs with minimum degree at least three. Lastly, we show that if the minimum degree is at least three then the girth of a WTD graph is at most 12. We conclude with several open questions.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73759102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-03DOI: 10.26493/1855-3974.1996.DB7
Tim Penttila, A. Siciliano
By using elementary linear algebra methods we exploit properties of the incidence map of certain incidence structures with finite block sizes. We give new and simple proofs of theorems of Kantor and Lehrer, and their infinitary version. Similar results are obtained also for diagrams geometries. By mean of an extension of Block’s Lemma on the number of orbits of an automorphism group of an incidence structure, we give informations on the number of orbits of: a permutation group (of possible infinite degree) on subsets of finite size; a collineation group of a projective and affine space (of possible infinite dimension) over a finite field on subspaces of finite dimension; a group of isometries of a classical polar space (of possible infinite rank) over a finite field on totally isotropic subspaces (or singular in case of orthogonal spaces) of finite dimension. Furthermore, when the structure is finite and the associated incidence matrix has full rank, we give an alternative proof of a result of Camina and Siemons. We then deduce that certain families of incidence structures have no sharply transitive sets of automorphisms acting on blocks.
{"title":"On the incidence maps of incidence structures","authors":"Tim Penttila, A. Siciliano","doi":"10.26493/1855-3974.1996.DB7","DOIUrl":"https://doi.org/10.26493/1855-3974.1996.DB7","url":null,"abstract":"By using elementary linear algebra methods we exploit properties of the incidence map of certain incidence structures with finite block sizes. We give new and simple proofs of theorems of Kantor and Lehrer, and their infinitary version. Similar results are obtained also for diagrams geometries. By mean of an extension of Block’s Lemma on the number of orbits of an automorphism group of an incidence structure, we give informations on the number of orbits of: a permutation group (of possible infinite degree) on subsets of finite size; a collineation group of a projective and affine space (of possible infinite dimension) over a finite field on subspaces of finite dimension; a group of isometries of a classical polar space (of possible infinite rank) over a finite field on totally isotropic subspaces (or singular in case of orthogonal spaces) of finite dimension. Furthermore, when the structure is finite and the associated incidence matrix has full rank, we give an alternative proof of a result of Camina and Siemons. We then deduce that certain families of incidence structures have no sharply transitive sets of automorphisms acting on blocks.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83637655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-24DOI: 10.26493/1855-3974.1852.4f7
W. Imrich, R. Kalinowski, M. Pilsniak, M. Wozniak
We consider edge colourings, not necessarily proper. The distinguishing index D ′( G ) of a graph G is the least number of colours in an edge colouring that is preserved only by the identity automorphism. It is known that D ′( G ) ≤ Δ for every countable, connected graph G with finite maximum degree Δ except for three small cycles. We prove that D ′( G ) ≤ ⌈√Δ⌉ + 1 if additionally G does not have pendant edges.
我们考虑边缘颜色，不一定是正确的。图G的区分指标D ' (G)是仅靠恒等自同构保持的边着色中颜色的最少个数。已知除三个小环外，对于最大有限次的可数连通图G Δ， D ' (G)≤Δ。我们证明了如果另外的G没有垂边，D ' (G)≤≤≤≤Δ²+ 1。
{"title":"The distinguishing index of connected graphs without pendant edges","authors":"W. Imrich, R. Kalinowski, M. Pilsniak, M. Wozniak","doi":"10.26493/1855-3974.1852.4f7","DOIUrl":"https://doi.org/10.26493/1855-3974.1852.4f7","url":null,"abstract":"We consider edge colourings, not necessarily proper. The distinguishing index D ′( G ) of a graph G is the least number of colours in an edge colouring that is preserved only by the identity automorphism. It is known that D ′( G ) ≤ Δ for every countable, connected graph G with finite maximum degree Δ except for three small cycles. We prove that D ′( G ) ≤ ⌈√Δ⌉ + 1 if additionally G does not have pendant edges.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80827857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-24DOI: 10.26493/1855-3974.1933.2DF
Z. Stanić
A connected signed graph is called exceptional if it has a representation in the root system E 8 , but has not in any D k . In this study we obtain some properties of these signed graphs, mostly expressed in terms of those that are maximal with a fixed number of eigenvalues distinct from −2 . As an application, we characterize exceptional signed graphs with exactly 2 eigenvalues. In some particular cases, we prove the (non-)existence of such signed graphs.
{"title":"Notes on exceptional signed graphs","authors":"Z. Stanić","doi":"10.26493/1855-3974.1933.2DF","DOIUrl":"https://doi.org/10.26493/1855-3974.1933.2DF","url":null,"abstract":"A connected signed graph is called exceptional if it has a representation in the root system E 8 , but has not in any D k . In this study we obtain some properties of these signed graphs, mostly expressed in terms of those that are maximal with a fixed number of eigenvalues distinct from −2 . As an application, we characterize exceptional signed graphs with exactly 2 eigenvalues. In some particular cases, we prove the (non-)existence of such signed graphs.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75841855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-10DOI: 10.26493/1855-3974.2405.B43
Gang Chen, Jiawei He, Ilia N. Ponomarenko, A. Vasil’ev
Recent classification of $frac{3}{2}$-transitive permutation groups leaves us with three infinite families of groups which are neither $2$-transitive, nor Frobenius, nor one-dimensional affine. The groups of the first two families correspond to special actions of ${mathrm{PSL}}(2,q)$ and ${mathrm{PGamma L}}(2,q),$ whereas those of the third family are the affine solvable subgroups of ${mathrm{AGL}}(2,q)$ found by D. Passman in 1967. The association schemes of the groups in each of these families are known to be pseudocyclic. It is proved that apart from three particular cases, each of these exceptional pseudocyclic schemes is characterized up to isomorphism by the tensor of its $3$-dimensional intersection numbers.
{"title":"A characterization of exceptional pseudocyclic association schemes by multidimensional intersection numbers","authors":"Gang Chen, Jiawei He, Ilia N. Ponomarenko, A. Vasil’ev","doi":"10.26493/1855-3974.2405.B43","DOIUrl":"https://doi.org/10.26493/1855-3974.2405.B43","url":null,"abstract":"Recent classification of $frac{3}{2}$-transitive permutation groups leaves us with three infinite families of groups which are neither $2$-transitive, nor Frobenius, nor one-dimensional affine. The groups of the first two families correspond to special actions of ${mathrm{PSL}}(2,q)$ and ${mathrm{PGamma L}}(2,q),$ whereas those of the third family are the affine solvable subgroups of ${mathrm{AGL}}(2,q)$ found by D. Passman in 1967. The association schemes of the groups in each of these families are known to be pseudocyclic. It is proved that apart from three particular cases, each of these exceptional pseudocyclic schemes is characterized up to isomorphism by the tensor of its $3$-dimensional intersection numbers.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76996606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-01DOI: 10.26493/1855-3974.2443.02e
Ted Dobson, M. Muzychuk, Pablo Spiga
In this paper, we find a strong new restriction on the structure of CI-groups. We show that, if $R$ is a generalised dihedral group and if $R$ is a CI-group, then for every odd prime $p$ the Sylow $p$-subgroup of $R$ has order $p$, or $9$. Consequently, any CI-group with quotient a generalised dihedral group has the same restriction, that for every odd prime $p$ the Sylow $p$-subgroup of the group has order $p$, or $9$. We also give a counter example to the conjecture that every BCI-group is a CI-group.
{"title":"Generalized dihedral CI-groups","authors":"Ted Dobson, M. Muzychuk, Pablo Spiga","doi":"10.26493/1855-3974.2443.02e","DOIUrl":"https://doi.org/10.26493/1855-3974.2443.02e","url":null,"abstract":"In this paper, we find a strong new restriction on the structure of CI-groups. We show that, if $R$ is a generalised dihedral group and if $R$ is a CI-group, then for every odd prime $p$ the Sylow $p$-subgroup of $R$ has order $p$, or $9$. Consequently, any CI-group with quotient a generalised dihedral group has the same restriction, that for every odd prime $p$ the Sylow $p$-subgroup of the group has order $p$, or $9$. We also give a counter example to the conjecture that every BCI-group is a CI-group.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87062870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-29DOI: 10.26493/1855-3974.2450.1DC
S. Skresanov
A permutation group $G$ on $Omega$ is called a rank 3 group if it has precisely three orbits in its induced action on $Omega times Omega$. The largest permutation group on $Omega$ having the same orbits as $G$ on $Omega times Omega$ is called the 2-closure of $G$. A description of 2-closures of rank 3 groups is given. As a special case, it is proved that 2-closure of a primitive one-dimensional affine rank 3 permutation group of sufficiently large degree is also affine and one-dimensional.
{"title":"On 2-closures of rank 3 groups","authors":"S. Skresanov","doi":"10.26493/1855-3974.2450.1DC","DOIUrl":"https://doi.org/10.26493/1855-3974.2450.1DC","url":null,"abstract":"A permutation group $G$ on $Omega$ is called a rank 3 group if it has precisely three orbits in its induced action on $Omega times Omega$. The largest permutation group on $Omega$ having the same orbits as $G$ on $Omega times Omega$ is called the 2-closure of $G$. A description of 2-closures of rank 3 groups is given. As a special case, it is proved that 2-closure of a primitive one-dimensional affine rank 3 permutation group of sufficiently large degree is also affine and one-dimensional.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77116550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-03DOI: 10.26493/1855-3974.2374.9ff
M. Buratti, Douglas R Stinson
We prove new existence and nonexistence results for modular Golomb rulers in this paper. We completely determine which modular Golomb rulers of order $k$ exist, for all $kleq 11$, and we present a general existence result that holds for all $k geq 3$. We also derive new nonexistence results for infinite classes of modular Golomb rulers and related structures such as difference packings, optical orthogonal codes, cyclic Steiner systems and relative difference families.
{"title":"New results on modular Golomb rulers, optical orthogonal codes and related structures","authors":"M. Buratti, Douglas R Stinson","doi":"10.26493/1855-3974.2374.9ff","DOIUrl":"https://doi.org/10.26493/1855-3974.2374.9ff","url":null,"abstract":"We prove new existence and nonexistence results for modular Golomb rulers in this paper. We completely determine which modular Golomb rulers of order $k$ exist, for all $kleq 11$, and we present a general existence result that holds for all $k geq 3$. We also derive new nonexistence results for infinite classes of modular Golomb rulers and related structures such as difference packings, optical orthogonal codes, cyclic Steiner systems and relative difference families.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87586996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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