The b-symbol metric is a generalization of the Hamming metric. Linear codes in the b-symbol metric have been used in the read channel whose outputs consist of b consecutive symbols. The Griesmer bound outperforms the Singleton bound for${\mathbb {F}}_{q}$-linear codes in the Hamming metric, when q is fixed and the length is large enough. This scenario is also applicable in the b-symbol metric. Shi, Zhu, and Helleseth recently made a conjecture on cyclic codes in the b-symbol metric. In this paper, we present the b-symbol Griesmer bound for linear codes by concatenating linear codes and simplex codes. Based on cyclic codes and extended cyclic codes, we propose two families of distance-optimal linear codes with respect to the b-symbol Griesmer bound.

Carlet et al. showed that for$q\geq 3$, there exists a linear complementary pair of codes over${\mathbb F}_{q}$whose security parameter is as good as the minimum distance$d_{L}$of the best linear code with the same length and dimension. For binary codes, they proved that the security parameter of a linear complementary pair of codes is lower bounded by$d_{L}-1$. Choi et al. recently presented infinite families of binary linear complementary pairs which are optimal in the sense that their security parameters reach$d_{L}$. Here, we prove that for every$k\geq 5$and$d\geq \lceil (k-1)/2 \rceil 2^{k-1}$, there exist binary linear complementary pairs of codes of length$g(k,d)$, where$g(k,d)$denotes the Griesmer bound. This shows the existence of an infinite family of optimal binary LCP of codes for new code parameters, which extensively broaden those obtained by Choi et al. Our construction is explicit and it is based on codes reaching the Griesmer bound, which were constructed by Solomon and Stiffler.

We study linear complementary pairs (LCP) of codes (C, D), where both codes belong to the same algebraic code family. We especially investigate constacyclic and quasicyclic LCP of codes. We obtain characterizations for LCP of constacyclic codes and LCP of quasi-cyclic codes. Our result for the constacyclic complementary pairs extends the characterization of linear complementary dual (LCD) cyclic codes given by Yang and Massey. We observe that when C and D are complementary and constacyclic, the codes C and D⊥are equivalent to each other. Hence, the security parameter min(d(C), d(D⊥)) for LCP of codes is simply determined by one of the codes in this case. The same holds for a special class of quasi-cyclic codes, namely 2D cyclic codes, but not in general for all quasi-cyclic codes, since we have examples of LCP of double circulant codes not satisfying this conclusion for the security parameter. We present examples of binary LCP of quasi-cyclic codes and obtain several codes with better parameters than known binary LCD codes. Finally, a linear programming bound is obtained for binary LCP of codes and a table of values from this bound is presented in the case d(C) = d(D⊥). This extends the linear programming bound for LCD codes.

We completely characterize possible indices of quasi-cyclic subcodes in a cyclic code for a very broad class of cyclic codes. We present enumeration results for quasi-cyclic subcodes of a fixed index and show that the problem of enumeration is equivalent to enumeration of certain vector subspaces in finite fields. In particular, we present enumeration results for quasi-cyclic subcodes of the simplex code and duals of certain Bose--Chaudhuri--Hocquenqhem codes. Our results are based on the trace representation of cyclic codes.

We introduce multidimensional analogues of quasi-cyclic (QC) codes and study their algebraic structure. We demonstrate a concatenated structure for multidimensional QC codes and use this to prove that this class of codes is asymptotically good. We also relate the new family of codes to convolutional codes. It is known that the minimum distance of QC codes provides a natural lower bound on the free distance of convolutional codes. We show that the same relation also holds between certain rank one 2-D convolutional codes and the related multidimensional QC codes. We provide examples, which show that our bound is sharp in some cases. We also present some optimal 2-D QC codes. Along the way, we provide a condition on the encoders of rank one convolutional codes, which are equivalent to noncatastrophicity for 1-D convolutional codes. In the nD case (n>1), our condition is sufficient for the noncatastrophicity of the encoder.

Following Jensen's work from 1985, a quasi-cyclic code can be written as a direct sum of concatenated codes, where the inner codes are minimal cyclic codes and the outer codes are linear codes. We observe that the outer codes are nothing but the constituents of the quasi-cyclic code in the sense of Ling-Solé. This concatenated structure enables us to recover some earlier results on quasi-cyclic codes in a simple way, including one of our recent results which says that a quasi-cyclic code with cyclic constituent codes are 2-D cyclic codes. In fact, we obtain a generalization of this result to multidimensional cyclic codes. The concatenated structure also yields a lower bound on the minimum distance of quasi-cyclic codes, as noted by Jensen, which we call Jensen's bound. We show that a recent lower bound on the minimum distance of quasi-cyclic codes that we obtained is in general better than Jensen's lower bound.

We give a general lower bound for the minimum distance of $q$-ary quasi-cyclic codes of length $m\ell$ and index $\ell$, where $m$ is relatively prime to $q$. The bound involves the minimum distances of constituent codes of length $\ell$ as well as the minimum distances of certain cyclic codes of length $m$ which are related to the fields over which the constituents are defined. We present examples which show that the bound is sharp in many instances. We also compare the performance of our bound against the bounds of Lally and Esmaeili-Yari.

We give a new method in order to obtain Weil-Serre type bounds on the minimum distance of arbitrary cyclic codes over${\BBF}_{p^e}$of length coprime to$p$, where$e \ge 1$is an arbitrary integer. In an earlier paper we obtained Weil-Serre type bounds for such codes only when$e=1$or$e=2$using lengthy explicit factorizations, which seems hopeless to generalize. The new method avoids such explicit factorizations and it produces an effective alternative. Using our method we obtain Weil–Serre type bounds in various cases. By examples we show that our bounds perform very well against Bose–Chaudhuri–Hocquenghem (BCH) bound and they yield the exact minimum distance in some cases.

We prove a Weil-Serre type bound on the number of solutions of a class of reducible additive equations over finite fields. Using the trace representation of cyclic codes, this enables us to write a general estimate for the weights of cyclic codes. We extend Wolfmann's weight bound to a larger classes of cyclic codes. In particular, our result is applicable to any cyclic code over Fpand Fp2, where p is an arbitrary prime. Examples indicate that our bound performs very well against the Bose-Chaudhuri-Hocquenghem (BCH) bound and that it yields the exact minimum distance in some cases