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Recently, the research of trace codes over finite fields has received a lot of attention because some of these codes have good parameters. In this paper, we construct several classes of trace codes from some special classes of linear codes and give their weight distributions, showing that they contain some optimal and almost optimal codes. Furthermore, we determine the parameters of the duals of the trace codes, and show that they are distance-optimal or nearly distance-optimal with respect to the sphere-packing bound. [ABSTRACT FROM AUTHOR]

Maximum-distance-separable (MDS) codes and self-dual codes are important families of classical codes in coding theory. It is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over finite field F q has been completely solved for q is even. In this paper, for finite field with odd characteristic, we obtain some new results on the existence of MDS self-dual codes by generalized Reed-Solomon (GRS) codes and extended GRS codes. [ABSTRACT FROM AUTHOR]

In this paper, we give the structures of all ( 1 + u + v ) -constacyclic codes of length 2 s over F 2 + u F 2 + v F 2 + u v F 2 , we also obtain the number of codewords, and the dual of each constacyclic code. Among other results, we study the form of such constacyclic codes that are self-dual and give the number of these codes. We end by giving all self-dual constacyclic codes of length 2, 4 and 8. [ABSTRACT FROM AUTHOR]

Let $$R=\mathbb {F}_{2^{m}}+u\mathbb {F}_{2^{m}}+\cdots +u^{k}\mathbb {F}_{2^{m}}$$ , where $$\mathbb {F}_{2^{m}}$$ is the finite field with $$2^{m}$$ elements, m is a positive integer, and u is an indeterminate with $$u^{k+1}=0.$$ In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over R. A new Gray map over R is defined, and a sufficient and necessary condition for the existence of dual-containing cyclic codes over R is given. A new family of $$2^{m}$$ -ary quantum codes is obtained via the Gray map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over R. In particular, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over R. [ABSTRACT FROM AUTHOR]

Let q be a power of an odd prime, and be the trace mapping. A mapping is called planar (or perfect nonlinear) on if for any non-zero , the difference mapping is a permutation where for , . Kyureghyan and Özbudak (2012) [8] considered the planarity of mappings on for and proved that there is no planar for . For the case and , they raised three conjectures. In this paper we prove the third conjecture which says that there is no planar for , by using Kloosterman sums. Our proof also works for case , so we present a new proof of the Kyureghyan–Özbudak result. For case , we present an elementary proof of the first conjecture which says that there is no planar for . [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum error-correcting (EAQEC) codes can be obtained from arbitrary classical linear codes, based on the entanglement-assisted stabilizer formalism. However, how to determine the required number of shared pairs is challenging. In this paper, we first construct three classes of classical linear MDS codes over finite fields by considering generalized Reed–Solomon codes and calculate the dimension of their Hermitian hulls. By using these MDS codes, we then obtain three new classes of EAQEC codes and EAQEC MDS codes, whose maximally entangled states can take various values. Moreover, these EAQEC codes have more flexible lengths. [ABSTRACT FROM AUTHOR]

Any permutation polynomial is an n -cycle permutation. When n is a specific small positive integer, one can obtain efficient permutations, such as involutions, triple-cycle permutations and quadruple-cycle permutations. These permutations have important applications in cryptography and coding theory. Inspired by the AGW Criterion, we propose criteria for n -cycle permutations, which mainly are of the form x r h (x s). We then propose unified constructing methods including recursive ways and a cyclotomic way for n -cycle permutations of such form. We demonstrate our approaches by constructing three classes of explicit triple-cycle permutations with high index and two classes of n -cycle permutations with low index, many of which are new both at levels of permutation property and cycle property. [ABSTRACT FROM AUTHOR]

Let R be the polynomial residue ring F q 2 + u F q 2 , where F q 2 is the finite field with q 2 elements, q is a power of a prime p, and u is an indeterminate with u 2 = 0. We introduce a Gray map from R to F q 2 p and study (1 - u) -constacyclic codes over R. It is proved that the image of a (1 - u) -constacyclic code of length n over R under the Gray map is a distance-invariant linear cyclic code of length pn over F q 2. We give some necessary and sufficient conditions for (1 - u) -constacyclic codes over R to be Hermitian dual-containing. In particular, a new class of 2 m -ary quantum codes is obtained via the Gray map and the Hermitian construction from Hermitian dual-containing (1 - u) -constacyclic codes over the ring F 2 2 m + u F 2 2 m . [ABSTRACT FROM AUTHOR]

In this paper, we explicitly determine the generator polynomials of all repeated-root constacyclic codes of length n l p s over F q and their dual codes, where l is an odd prime different from p , and n is an odd prime different from both l and p such that n = 2 h + 1 for some prime h or h = 1 . Moreover, we give all self-dual cyclic codes of length n l p s over F q and their enumeration. All linear complementary dual (LCD) cyclic codes of length n l p s over F q and their enumeration are also obtained. [ABSTRACT FROM AUTHOR]