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A linear code is linear complementary dual (LCD) if it meets its dual trivially. LCD codes have been a hot topic recently due to Boolean masking application in the security of embarked electronics (Carlet and Guilley in Pinto et al (eds) Coding theory and applications, Springer, CIMSMS, Berlin, 2015). Additive codes over F 4 are F 4 -codes that are stable by codeword addition but not necessarily by scalar multiplication. An additive code over F 4 is additive complementary dual (ACD) if it meets its dual trivially. The aim of this research is to study such codes which meet their dual trivially. All the techniques and problems used to study LCD codes are potentially relevant to ACD codes. Interesting constructions of ACD codes from binary codes are given with respect to the trace Hermitian and trace Euclidean inner product. The former product is relevant to quantum codes. [ABSTRACT FROM AUTHOR]

This paper investigates the existence, enumeration, and asymptotic performance of self-dual and LCD double circulant codes over Galois rings of characteristic p2 and order p4 with p an odd prime. When p ≡ 3 (mod 4), we give a method to construct a duality preserving bijective Gray map from such a Galois ring to Z2p2. Closed formed enumeration formulas for double circulant codes that are self-dual (resp. LCD) are derived as a function of the length of these codes. Using random coding, we obtain families of asymptotically good self-dual and LCD codes over Zp2 with respect to the metric induced by the standard Fp-valued Gray maps. [ABSTRACT FROM AUTHOR]

We study the asymptotic performance of quasi-twisted codes viewed as modules in the ring R=Fq[x]/⟨xn+1⟩, when they are self-dual and of length 2n or 4n. In particular, in order for the decomposition to be amenable to analysis, we study factorizations of xn+1 over Fq, with n twice an odd prime, containing only three irreducible factors, all self-reciprocal. We give arithmetic conditions bearing on n and q for this to happen. Given a fixed q,  we show these conditions are met for infinitely many n’s, provided a refinement of Artin primitive root conjecture holds. This number theory conjecture is known to hold under generalized Riemann hypothesis (GRH). We derive a modified Varshamov-Gilbert bound on the relative distance of the codes considered, building on exact enumeration results for given n and q. [ABSTRACT FROM AUTHOR]

Four circulant codes form a special class of 2 -generator, index 4 , quasi-cyclic codes. Under some conditions on their generator matrix they can be shown to be self-dual. Artin primitive root conjecture shows the existence of an infinite subclass of these codes satisfying a modified Gilbert–Varshamov bound. [ABSTRACT FROM AUTHOR]

By a sphere-packing argument, we show that there are infinitely many pairs of primes that are close to each other for some metrics on the integers. In particular, for any numeration basis q, we show that there are infinitely many pairs of primes the base q expansion of which differ in at most two digits. Likewise, for any fixed integer t, there are infinitely many pairs of primes, the first t digits of which are the same. In another direction, we show that, there is a constant c depending on q such that for infinitely many integers m there are at least c log log m primes which differ from m by at most one base q digit. [ABSTRACT FROM AUTHOR]

In this paper, we solve the open problems raised in [8] and present some examples to illustrate the obtained results. Moreover, we work out the diophantine problem by Shi and Wang and then give the sufficient conditions for the nonexistence of two-Lee weight projective codes over \mathbb Z4 with type 4^k12^{k2} . [ABSTRACT FROM PUBLISHER]

Constacyclic codes are an important class of linear codes in coding theory. Many optimal linear codes are directly derived from constacyclic codes. In this paper, (1 + u)-constacyclic codes over Z + uZ of any length are studied. A new Gray map between Z + uZ and Z is defined. By means of this map, it is shown that the Z Gray image of a (1 + u)-constacyclic code of length n over Z + uZ is a cyclic code over Z of length 4 n. Furthermore, by combining the classical Gray map between Z and F , it is shown that the binary image of a (1 + u)-constacyclic code of length n over Z + uZ is a distance invariant binary quasi-cyclic code of index 4 and length 8 n. Examples of good binary codes are constructed to illustrate the application of this class of codes. [ABSTRACT FROM AUTHOR]