Linear codes are widely employed in communication systems, consumer electronics, and storage devices. All linear codes over finite fields can be generated by a generator matrix. Due to this, the generator matrix approach is called a fundamental construction of linear codes. This is the only known construction method that can produce all linear codes over finite fields. Recently, a defining-set ...

In this paper, we show that LCD codes are not equivalent to linear codes over small finite fields. The enumeration of binary optimal LCD codes is obtained. We also get the exact value of LD(n, 2) over F3 and F4. We study the bound of LCD codes over Fq.

Self-orthogonal codes have been of interest due to there rich algebraic structures and wide applications. Euclidean self-orthogonal codes have been quite well studied in literature. Here, we have focused on Hermitian self-orthogonal codes. Constructions of such codes have been given based on the well-known matrix-product construction for linear codes. Criterion for the underlying matrix and the...

Let nq(k, d) be the smallest integer n for which there exists an [n, k, d]q code for given q, k, d. It is known that n8(4, d) = ∑3 i=0 ⌈ d/8i ⌉ for all d ≥ 833. As a continuation of Jones et al. [Electronic J. Combinatorics 13 (2006), #R43], we determine n8(4, d) for 117 values of d with 113 ≤ d ≤ 832 and give upper and lower bounds on n8(4, d) for other d using geometric methods and some exten...

Let P = n11 ⊕ · · · ⊕ nt1 be the poset given by the ordinal sum of the antichains ni1 with ni elements. We derive MacWilliams-type identities for the fragment and sphere enumerators, relating enumerators for the dual C of the linear code C on P and those for C on the dual poset P̌. The linear changes of variables appearing in the identities are explicit. So we obtain, for example, the P-weight d...

This paper gives the minimum generating sets of three types of one generator (1 + u)-quasi twisted (QT) codes over F 2 + uF 2 , u 2 = 0. Moreover, it discusses the generating sets and the lower bounds on the minimum Lee distance of a special class of A 2 type one generator (1 + u)-QT codes. Some good (optimal or suboptimal) linear codes over F 2 are obtained by these types of one generator (1 +...

Consider binary linear codes obtained from bipartite graphs as follows. There are k ≥ 1 left nodes each representing a message bit and there are m = m(k) right nodes each representing a parity bit, generated from the corresponding set of message node neighbours. Both the message and the parity bits are sent through a memoryless binary input channel that either retains, flips or erases each tran...

For a field F, let R(n,m) be the ring of invariant polynomials for the action of SL(n,F) × SL(n,F) on tuples of matrices – (A,C) ∈ SL(n,F)× SL(n,F) sends (B1, . . . , Bm) ∈ M(n,F) to (AB1C , . . . , ABmC ). In this paper we call R(n,m) the ring of matrix semi-invariants. Let β(R(n,m)) be the smallest D s.t. matrix semi-invariants of degree ≤ D generate R(n,m). Guided by the Procesi-Razmyslov-Fo...

We will show how one can compute all reduced Gröbner bases with respect to a degree compatible ordering for code ideals even though these binomial ideals are not toric. To this end, the correspondence of linear codes and binomial ideals will be briefly described as well as their resemblance to toric ideals. Finally, we will hint at applications of the degree compatible Gröbner fan to the code e...