The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field${\mathrm {GF}}(q)$with length$q+1$, which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some$q$-ary BCH codes with length$q+1$, and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of${\mathrm {PGL}}(2, p^{m})$-invariant codes over${\mathrm {GF}}(p^{h})$is completed. As an application of this result, the$p$-ranks of all incidence structures invariant under the projective general linear group${\mathrm {PGL}}(2, p^{m})$are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms.

Bent functions, or equivalently, Hadamard difference sets in the elementary Abelian group (GF(22m), +), have been employed to construct symmetric and quasi-symmetric designs having the symmetric difference property. The main objective of this paper is to use bent vectorial functions for a construction of a two-parameter family of binary linear codes that do not satisfy the conditions of the Assmus-Mattson theorem, but nevertheless hold 2-designs. A new coding-theoretic characterization of bent vectorial functions is presented.

The projective special linear group PSL2(n) is 2-transitive for all primes n and 3-homogeneous for n = 3 (mod 4) on the set (0, 1, ... , n - 1, ∞). It is known that the extended odd-like quadratic residue codes are invariant under PSL2(n). Hence, the extended quadratic residue codes hold an infinite family of 2-designs for primes n = 1 (mod 4), an infinite family of 3-designs for primes n = 3 (mod 4). To construct more t-designs with t ∈ (2, 31, one would search for other extended cyclic codes over finite fields that are invariant under the action of PSL2(n). The objective of this paper is to prove that the extended quadratic residue binary codes are the only nontrivial extended binary cyclic codes that are invariant under PSL2(n).

Self-synchronization under the presence of additive noise can be achieved by allocating a certain number of bits of each codeword as markers for synchronization. Difference systems of sets are combinatorial designs which specify the positions of synchronization markers in codewords in such a way that the resulting error-tolerant self-synchronizing codes may be realized as cosets of linear codes. Ideally, difference systems of sets should sacrifice as few bits as possible for a given code length, alphabet size, and error-tolerance capability. However, it seems difficult to attain optimality with respect to known bounds when the noise level is relatively low. In fact, the majority of known optimal difference systems of sets are for exceptionally noisy channels, requiring a substantial amount of bits for synchronization. To address this problem, we present constructions for difference systems of sets that allow for higher information rates while sacrificing optimality to only a small extent. Our constructions utilize optimal difference systems of sets as ingredients and, when applied carefully, generate asymptotically optimal ones with higher information rates. We also give direct constructions for optimal difference systems of sets with high information rates and error tolerance that generate binary and ternary self-synchronizing codes.

As in classical coding theory, quantum analogs of low-density parity-check (LDPC) codes have offered good error correction performance and low decoding complexity by employing the Calderbank-Shor-Steane construction. However, special requirements in the quantum setting severely limit the structures such quantum codes can have. While the entanglement-assisted stabilizer formalism overcomes this limitation by exploiting maximally entangled states (ebits), excessive reliance on ebits is a substantial obstacle to implementation. This paper gives necessary and sufficient conditions for the existence of quantum LDPC codes which are obtainable from pairs of identical LDPC codes and consume only one ebit, and studies the spectrum of attainable code parameters.

New improved upper and lower bounds on the maximum size of a symmetric or arbitrary conflict-avoiding code of length n = 4 m for three active users are proved. Furthermore, direct constructions for optimal conflict-avoiding codes of length n = 4 m and m equiv 2 (mod 4) for three active users are provided.

The paper surveys some recent combinatorial constructions of optimal comma-free codes being cosets of linear codes that explore cyclotomy and partitions of hyperplanes or complements of hyperplanes in a finite projective geometry.

We consider the problem to construct a code of the maximum cardinality which consists of binary vectors of length n with three ones and has the following property: a matrix of size 3 times n from any cyclic shifts of any three different code vectors contains the identity matrix of size 3 times 3 (with accuracy up to a permutation of columns). This property (in more general form) was considered in the connection with the problem to avoid conflicts in the channels of multiple access under a restriction to the number of active users (see L. A. Bassalygo and M. S. Pinsker, Problemy Peredachi Informatsii, 1983; J. L. Massey and P. Mathys, IEEE Trans. Inform. Theory , 1985; B.S. Tsybakov and A.R. Rubinov, Problems of Information Transmission , 2002). The cardinality of such a code corresponds to the number of all users, and this property means that each from any three active users can successfully transmit a packet of information in one of three attempts to do it during n slots of time without a collision with other active users. In particular, cyclic Steiner triple systems give examples of such conflict-avoiding codes if we choose representatives of the cyclic classes as code vectors. In the paper we present some constructions of conflict-avoiding codes of triples which are better as compared with those obtained from the cyclic Steiner triple systems

It is proved that a class of q-ary (2n,n) formally self-dual codes obtained from symmetric matrices over GF (q), contains codes that meet the Varshamov-Gilbert bound. The codes are self-dual with respect to the symplectic inner product and yield quantum codes encoding one state with n q-ary qubits and having minimum distance proportional to n.

The following lower bound for binary constant weight codes are derived by an explicit construction: A(17,4,5)/spl ges/441. The construction exploits maximal sets of bases in the four-dimensional binary vector space pairwise intersecting in at most two vectors.

Some binary linear codes of length 50 and 100 are constructed using the adjacency matrices of the Hoffman-Singleton graph and the Higman-Sims graph, Some of the codes are optimal or nearly optimal for the given length and dimension. The dual codes admit majority logic decoding.

Some new extremal binary [54,27,10] self-dual codes are constructed using automorphisms of order 7.

J.H. Conway and N.J.A. Sloane (see ibid., vol.36, no.6, p.1319-33, Nov. 1990) give weight enumerators of several self-dual codes with the highest possible minimal distance whose existence was not known. A generator matrix for one of these, a Type I (64, 32, 12) code, is given, proving its existence. The method of construction is described.>

It is shown that the covering radius of any binary linear (14, 6) code containing the all-one vector is at least 4. Since the minimum covering radius of a binary linear (14, 6) code is 3, this shows that in general the minimum of the covering radius is not achieved by codes containing the all-one vector.>

All maximal self-orthogonal and self-dual codes of lengthnwheren\leq 9are completely classified. A generator matrix, Hamming weight distribution, and order of the monomial group of each of these codes are given.

An asymptotic estimate is established for the number of equivalence classes of Boolean functions ofnvariables under transformations of the formf(x) \rightarrow f(Ax+b)+ L(x), wherex=(x_{l}, \cdots ,x_{n}), Ais a nonsingularnbynmatrix,bis a vector, andL(x)is a linear function.