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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 quasi-cyclic 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^\bot $ are equivalent to each other. Hence, the security parameter $\min (d(C),d(D^\bot))$ 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 $d(C)=d(D^\bot)$. This extends the linear programming bound for LCD codes. [ABSTRACT FROM AUTHOR]

The construction of deletion codes for the editing metric is reduced to the construction of codes over the integers for the Manhattan metric by run length coding. The latter codes are constructed by expurgation of lattices’ translates. These lattices, in turn, are obtained from Construction A applied to binary codes and \mathbb Z4 –codes. A lower bound on the size of our codes for the Manhattan distance are obtained through generalized theta series of the corresponding lattices. For any fixed number of deletions, provided the number of runs is large enough our method supplies a correction technique. For fixed number of runs and binary sequence length large our lattice construction is shown to be tight up to constants. [ABSTRACT FROM PUBLISHER]

We introduce the class of multiply constant-weight codes to improve the reliability of certain physically unclonable function response, and extend classical coding methods to construct multiply constant-weight codes from known \(q\) -ary and constant-weight codes. We derive analogs of Johnson bounds and give constructions showing these bounds to be asymptotically tight up to a constant factor under certain conditions. We also examine the rates of multiply constant-weight codes and demonstrate that these rates are the same as those of constant-weight codes of corresponding parameters. [ABSTRACT FROM PUBLISHER]

Hermitian self-dual abelian codes in a group ring \BBFq^{2}[G], where \BBFq^{2} is a finite field of order q^2 and G is a finite abelian group, are studied. Using the well-known discrete Fourier transform decomposition for a semisimple group ring, a characterization of Hermitian self-dual abelian codes in \BBFq^{2}[G] is given, together with an alternative proof of necessary and sufficient conditions for the existence of such a code in \BBFq^{2}[G], i.e., there exists a Hermitian self-dual abelian code in \BBFq^{2}[G] if and only if the order of G is even and q=2^l for some positive integer l. Later on, the study is further restricted to the case where \BBF2^{2l}[G] is a principal ideal group ring, or equivalently, G\cong A\oplus\BBZ2^{k} with 2\nmid\vert A\vert. Based on the characterization obtained, the number of Hermitian self-dual abelian codes in \BBF{2^{2l}}[A\oplus\BBZ2^{k}] can be determined easily. When A is cyclic, this result answers an open problem of Jia concerning Hermitian self-dual cyclic codes. In many cases, \BBF{2^{2l}}[A\oplus\BBZ2^{k}] contains a unique Hermitian self-dual abelian code. The criteria for such cases are determined in terms of l and the order of A. Finally, the distribution of finite abelian groups A such that a unique Hermitian self-dual abelian code exists in \BBF{2^{2l}}[A\oplus\BBZ2] is established, together with the distribution of odd integers m such that a unique Hermitian self-dual cyclic code of length 2 m over \BBF{2^{2l}} exists. [ABSTRACT FROM AUTHOR]

In this paper we classify all extremal and s-extremal binary self-dual codes of length 38. There are exactly 2744 extremal [38,19,8] self-dual codes, two s-extremal [38,19,6] codes, and 1730 s-extremal [38,19,8] codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self-dual codes of length 36, and from a generalization of this recursive algorithm for the shadow. The classification of s-extremal [38,19,6] codes permits to achieve the classification of all s-extremal codes with d=6. [ABSTRACT FROM AUTHOR]

The maximum size of a binary code is studied as a function of its length n, minimum distance d, and minimum codeword weight \ssi w. This function B(n,d,\ssi w) is first characterized in terms of its exponential growth rate in the limit n\rightarrow\infty for fixed \delta=d/n and \omega=\ssi w/n. The exponential growth rate of B(n,d,\ssi w) is shown to be equal to the exponential growth rate of A(n,d) for 0\leq\omega\leq 1/2, and equal to the exponential growth rate of A(n,d,\ssi w) for 1/2<\omega\leq 1. Second, analytic and numerical upper bounds on B(n,d,\ssi w) are derived using the semidefinite programming (SDP) method. These bounds yield a nonasymptotic improvement of the second Johnson bound and are tight for certain values of the parameters. [ABSTRACT FROM AUTHOR]

We present a general construction of asymmetric quantum codes based on additive codes under the trace Hermitian inner product. Various families of additive codes over \BBF 4 are used in the construction of many asymmetric quantum codes over \BBF 4. [ABSTRACT FROM PUBLISHER]

The weights in maximum distance separable (MDS) codes of length n and dimension k over the finite field GF(q) are studied. Up to some explicit exceptional cases, the MDS codes with parameters given by the MDS conjecture are shown to contain all k weights in the range n-k+1 to n. The proof uses the covering radius of the dual code. [ABSTRACT FROM AUTHOR]

We study binary self-dual codes with a fixed point free automorphism of order three. All binary codes of that type can be obtained by a cubic construction that generalizes Turyn's. We regard such "cubic" codes of length 3l as codes of length l over the ring F[sub 2] × F[sub 4]. Classical notions of Type II codes, shadow codes, and weight enumerators are adapted to that ring. Two infinite families of cubic codes are introduced. New extremal binary codes in lengths ≤ 66 are constructed by a randomized algorithm. Necessary conditions for the existence of a cubic [72, 36, 16] Type II code are derived. [ABSTRACT FROM AUTHOR]

Presents information on a study which described an algebraic approach to quasi-cyclic codes. Some basic notations and definitions; Trace formula for quasi-cyclic codes; Conclusions.

Focuses on a study which investigated duadic codes which constitute a family of binary cyclic codes. Interest in codes over finite rings; Theorems and proofs on duadic codes; Augmented and extended Abelian codes.