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Boolean functions, coding theory and $t$ -designs have close connections and interesting interplay. A standard approach to constructing $t$ -designs is the use of linear codes with certain regularity. The Assmus-Mattson Theorem and the automorphism groups are two ways for proving that a code has sufficient regularity for supporting $t$ -designs. However, some linear codes hold $t$ -designs, although they do not satisfy the conditions in the Assmus-Mattson Theorem and do not admit a $t$ -transitive or $t$ -homogeneous group as a subgroup of their automorphisms. The major objective of this paper is to develop a theory for explaining such codes and obtaining such new codes and hence new $t$ -designs. To this end, a general theory for punctured and shortened codes of linear codes supporting $t$ -designs is established, a generalized Assmus-Mattson theorem is developed, and a link between 2-designs and differentially $\delta $ -uniform functions and 2-designs is built. With these general results, binary codes with new parameters and explicit weight distributions are obtained, new 2-designs and Steiner system $S(2, 4, 2^{n})$ are produced in this paper. [ABSTRACT FROM AUTHOR]

Fractional repetition (FR) codes are a class of regenerating codes for distributed storage systems with an exact (table-based) repair process that is also uncoded, i.e., upon failure, a node is regenerated by simply downloading packets from the surviving nodes. In this paper, we present the constructions of FR codes based on Steiner systems and resolvable combinatorial designs, such as affine geometries, Hadamard designs, and mutually orthogonal Latin squares. The failure resilience of our codes can be varied in a simple manner. We construct codes with normalized repair bandwidth ( $\beta $ ) strictly larger than one; these cannot be obtained trivially from codes with $\beta = 1$ . Furthermore, we present the Kronecker product technique for generating new codes from existing ones and elaborate on their properties. FR codes with locality are those where the repair degree is smaller than the number of nodes contacted for reconstructing the stored file. For these codes, we establish a tradeoff between the local repair property and the failure resilience and construct codes that meet this tradeoff. Much of prior work only provided lower bounds on the FR code rate. In this paper, for most of our constructions, we determine the code rate for certain parameter ranges. [ABSTRACT FROM AUTHOR]

Constant-weight codes play an important role in coding theory. The problem of determining the sizes for optimal quaternary constant-weight codes with length n, weight 4, and minimum Hamming distance 5 ((n,5,4)4 codes) has been investigated in several papers. Although some constructions and several infinite families for such codes with length n\equiv 0,1\pmod4 have been given, the problem is still far from complete. In this paper, we determine the size of an optimal (n,5,4)4 code for each integer n\geq 4 leaving 55 lengths unsolved. Especially, for length n\equiv 0,1\pmod 4, the existence problem of the equivalent combinatorial object, namely the generalized Steiner system, is solved leaving only seven values undetermined. [ABSTRACT FROM AUTHOR]

An optimal constant-composition or constant-weight code of weight w has linear size if and only if its distance d is at least 2w - 1. When d ≥ 2w, the determination of the exact size of such a constant-composition or constant-weight code is trivial, but the case of d = 2w - 1 has been solved previously only for binary and ternary constant-composition and constant-weight codes, and for some sporadic instances. This paper provides a construction for quasicyclic optimal constant-composition and constant-weight codes of weight w and distance 2w - 1 based on a new generalization of difference triangle sets. As a result, the sizes of optimal constant-composition codes and optimal constant-weight codes of weight w and distance 2w - 1 are determined for all such codes of sufficiently large lengths. This solves an open problem of Etzion. The sizes of optimal constant-composition codes of weight w and distance 2w - 1 are also determined for all to ≤ 6, except in two cases. [ABSTRACT FROM AUTHOR]

A complete classification of the perfect binary one-error-correcting codes of length 15 as well as their extensions of length 16 is presented. There are 5983 such inequivalent perfect codes and 2165 extended perfect codes. Efficient generation of these codes relies on the recent classification of Steiner quadruple systems of order 16. Utilizing a result of Blackmore, the optimal binary one-error-correcting codes of length 14 and the (15, 1024, 4) codes are also classified; there are 38 408 and 5983 such codes, respectively. [ABSTRACT FROM AUTHOR]

In this correspondence, we study the probability of undetected error for binary constant weight codes. First, we derive a new lower bound on the probability of undetected error for binary constant weight codes. Next, we show that this bound is tight if and only if the binary constant weight codes are generated from certain t-designs in combinatorial design theory. This means that these binary constant weight codes generated from certain t-designs are uniformly optimal for error detection. Along the way, we determine the distance distributions of such binary constant weight codes. In particular, it is shown that binary constant weight codes generated from Steiner systems are uniformly optimal for error detection. Thus, we prove a conjecture of Xia, Fu, Jiang, and Ling. Furthermore, the distance distribution of a binary constant weight code generated from a Steiner system is determined. Finally, we study the exponent of the probability of undetected error for binary constant weight codes. We derive some bounds on the exponent of the probability of undetected error for binary constant weight codes. These bounds enable us to extend the region in which the exponent of the probability of undetected error is exactly determined. [ABSTRACT FROM AUTHOR]

Considers perfect single error-correcting codes or one-perfect codes. Definitions and properties about Steiner triple systems and propelinear codes; Main results about 1-perfect additive codes; Description of coding and decoding algorithms.