On nonsystematic perfect codes over finite fields

The nonsystematic perfect q-ary codes over finite field F _q of length n = (q ^m − 1)/(q − 1) are constructed in the case when m ≥ 4 and q ≥ 2 and also when m = 3 and q is not prime. For q ≠ 3, 5, these codes can be constructed by switching seven disjoint components of the Hamming code H _q ⁿ ; and, for q = 3, 5, eight disjoint components.     

On quasi-twisted codes over finite fields

In coding theory, quasi-twisted (QT) codes form an important class of codes which has been extensively studied. We decompose a QT code to a direct sum of component codes – linear codes over rings. Furthermore, given the decomposition of a QT code, we can describe the decomposition of its dual code. We also use the generalized discrete Fourier transform to give the inverse formula for both the nonrepeated-root and repeated-root cases. Then we produce a formula which can be used to construct a QT code given the component codes.     

On the locality of quasi-cyclic codes over finite fields

Designs, Codes and Cryptography - A code is said to have locality r if any coordinate value in a codeword of that code can be recovered by at most r other coordinates. In this paper, we have...     

Galois LCD codes over finite fields

In this paper, we generalize the linear complementary dual codes (LCD codes for short) to k-Galois LCD codes, and study them by a uniform method. A necessary and sufficient condition for linear codes to be k-Galois LCD codes is obtained, two classes of k-Galois LCD MDS codes are exhibited. Then, necessary and sufficient conditions for λ-constacyclic codes being k-Galois LCD codes are characterized. Some classes of k-Galois LCD λ-constacyclic MDS codes are constructed. Finally, we study Hermitian LCD λ-constacyclic codes, and present a class of Hermitian LCD λ-constacyclic MDS codes.     

New MDS self-dual codes over finite fields

In this paper we construct MDS Euclidean and Hermitian self-dual codes which are extended cyclic duadic codes or negacyclic codes. We also construct Euclidean self-dual codes which are extended negacyclic codes. Based on these constructions, a large number of new MDS self-dual codes are given with parameters for which self-dual codes were not previously known to exist.     

Quadratic spaces over finite fields and codes

Let V be a finite-dimensional quadratic space over a finite field GF(?) of characteristic different from 2. It is shown that, even if V is singular, the geometry of V is completely determined by the number of points on the unit sphere, the “sphere of the nonsquares,” and the “0-sphere.” For ? = 3, this implies that two codes over GF(3) with the same weight enumerator are isometric.     

Some properties of skew codes over finite fields

After recalling the definition of some codes as modules over skew polynomial rings, whose multiplication is defined by using an endomorphism and a derivation, and some basic facts about them, in the first part of this paper we study some of their main algebraic and geometric properties. Finally, for module skew codes constructed only with an automorphism, we give some BCH type lower bounds for their minimum distance.     

Extension theorems for linear codes over finite fields

We survey recent results on the extendability of linear codes over finite fields with link to projective geometry and some applications to optimal linear codes problem.     

A class of minimal cyclic codes over finite fields

Let ${\mathbb{F}}_q$ be a finite field of odd order $q$ and $n=2^a{p}_1^{a_1}{p}_2^{a_2}$, where $a,a_1,a_2$ are positive integers, $p_1,p_2$ are distinct odd primes and $4p_1{p}_2|q?1$. In this paper, we study the irreducible factorization of $x^n?1$ over ${\mathbb{F}}_q$ and all primitive idempotents in the ring ${\mathbb{F}}_q\left[x\right]∕〈x^n?1〉$.Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length $n$ over ${\mathbb{F}}_q$.