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Register a new userConstruction $C^star$ from Self-Dual Codes
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Added by
Maiara F. Bollauf
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Construction $C^star$ was recently introduced as a generalization of the multilevel Construction C (or Forneys code-formula), such that the coded levels may be dependent. Both constructions do not produce a lattice in general, hence the central idea of this paper is to present a 3-level lattice Construction $C^star$ scheme that admits an efficient nearest-neighborhood decoding. In order to achieve this objective, we choose coupled codes for levels 1 and 3, and set the second level code C2 as an independent linear binary self-dual code, which is known to have a rich mathematical structure among families of linear codes. Our main result states a necessary and sufficient condition for this construction to generate a lattice. We then present examples of efficient lattices and also non-lattice constellations with good packing properties.
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The parameters of a $q$-ary MDS Euclidean self-dual codes are completely determined by its length and the construction of MDS Euclidean self-dual codes with new length has been widely investigated in recent years. In this paper, we give a further study on the construction of MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. The main idea of our construction is to choose suitable evaluation points such that the corresponding (extended) GRS codes are Euclidean self-dual. Firstly, we consider the evaluation set consists of two disjoint subsets, one of which is based on the trace function, the other one is a union of a subspace and its cosets. Then four new families of MDS Euclidean self-dual codes are constructed. Secondly, we give a simple but useful lemma to ensure that the symmetric difference of two intersecting subsets of finite fields can be taken as the desired evaluation set. Based on this lemma, we generalize our first construction and provide two new families of MDS Euclidean self-dual codes. Finally, by using two multiplicative subgroups and their cosets which have nonempty intersection, we present three generic constructions of MDS Euclidean self-dual codes with flexible parameters. Several new families of MDS Euclidean self-dual codes are explicitly constructed.
Self-dual codes over $Z_2timesZ_4$ are subgroups of $Z_2^alpha timesZ_4^beta$ that are equal to their orthogonal under an inner-product that relates to the binary Hamming scheme. Three types of self-dual codes are defined. For each type, the possible values $alpha,beta$ such that there exist a code $Csubseteq Z_2^alpha timesZ_4^beta$ are established. Moreover, the construction of a $add$-linear code for each type and possible pair $(alpha,beta)$ is given. Finally, the standard techniques of invariant theory are applied to describe the weight enumerators for each type.
Systematic constructions of MDS self-dual codes is widely concerned. In this paper, we consider the constructions of MDS Euclidean self-dual codes from short length. Indeed, the exact constructions of MDS Euclidean self-dual codes from short length ($n=3,4,5,6$) are given. In general, we construct more new of $q$-ary MDS Euclidean self-dual codes from MDS self-dual codes of known length via generalized Reed-Solomon (GRS for short) codes and extended GRS codes.
Many $q$-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal $q^2$-ary linear codes. This result can be generalized to $q^{2 m}$-ary linear codes, $m > 1$. We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to cite{codet} and new $q$-ary ones, with $q eq 2$, improving others in the literature.
The parameters of MDS self-dual codes are completely determined by the code length. In this paper, we utilize generalized Reed-Solomon (GRS) codes and extended GRS codes to construct MDS self-dual (self-orthogonal) codes and MDS almost self-dual codes over. The main idea of our constructions is to choose suitable evaluation points such that the corresponding (extended) GRS codes are Euclidean self-dual (self-orthogonal). The evaluation sets are consists of two subsets which satisfy some certain conditions and the length of these codes can be expressed as a linear combination of two factors of q-1. Four families of MDS self-dual codes, two families of MDS self-orthogonal codes and two families of MDS almost self-dual codes are obtained and they have new parameters.
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