In this paper, we study the minimum distances of binary linear codes with parity check matrices formed from subset inclusion matrices , representing -element subsets versus -element subsets of an -element set. We provide both lower and upper bounds on the minimum distances of these codes and determine the exact values for any and sufficiently large . Our study combines design and integer linear programming techniques. The codes we consider are connected to LDPC codes and combinatorial designs. Furthermore, we construct quasi-cyclic LDPC codes from inclusion matrices that exhibit performance comparable to or slightly better than MacKay-type codes when evaluated using bit flipping and min-sum algorithms.
We provide some difference triangle sets with scopes that improve upon the best known values. These are found with purpose-built digital circuits realized with field-programmable gate arrays (FPGAs) rather than software algorithms running on general-purpose processors.
Let be an array whose top left sub-array is filled with a set of different symbols such that each cell of contains symbols. In this note, we find conditions under which each empty cell of can be filled with symbols in such a way that the total number of occurrences of each symbol is prescribed and that each symbol occurs at most
This paper investigates five kinds of systems of -partitions of , including symmetric Bollobás systems, strong Bollobás systems, Bollobás systems, skew Bollobás systems, and weak Bollobás systems. Many known results on variations of Bollobás systems are unified. Especially, we give a negative answer to a conjecture on Bollobás systems of -partitions of that was presented by Hegedüs and Frankl (European J. Comb., 120 [2024], 103983). Even though this conjecture does not hold for general Bollobás systems, we show that it holds for strong Bollobás systems of -partitions of .
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