اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدConstruction and enumeration of left dihedral codes satisfying certain duality properties
183
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Let $mathbb{F}_{q}$ be the finite field of $q$ elements and let $D_{2n}=langle x,ymid x^n=1, y^2=1, yxy=x^{n-1}rangle$ be the dihedral group of order $n$. Left ideals of the group algebra $mathbb{F}_{q}[D_{2n}]$ are known as left dihedral codes over $mathbb{F}_{q}$ of length $2n$, and abbreviated as left $D_{2n}$-codes. Let ${rm gcd}(n,q)=1$. In this paper, we give an explicit representation for the Euclidean hull of every left $D_{2n}$-code over $mathbb{F}_{q}$. On this basis, we determine all distinct Euclidean LCD codes and Euclidean self-orthogonal codes which are left $D_{2n}$-codes over $mathbb{F}_{q}$. In particular, we provide an explicit representation and a precise enumeration for these two subclasses of left $D_{2n}$-codes and self-dual left $D_{2n}$-codes, respectively. Moreover, we give a direct and simple method for determining the encoder (generator matrix) of any left $D_{2n}$-code over $mathbb{F}_{q}$, and present several numerical examples to illustrative our applications.
قيم البحث
اقرأ أيضاً
The well known Plotkin construction is, in the current paper, generalized and used to yield new families of Z2Z4-additive codes, whose length, dimension as well as minimum distance are studied. These new constructions enable us to obtain families of
Z2Z4-additive codes such that, under the Gray map, the corresponding binary codes have the same parameters and properties as the usual binary linear Reed-Muller codes. Moreover, the first family is the usual binary linear Reed-Muller family.
New quaternary Plotkin constructions are given and are used to obtain new families of quaternary codes. The parameters of the obtained codes, such as the length, the dimension and the minimum distance are studied. Using these constructions new famili
es of quaternary Reed-Muller codes are built with the peculiarity that after using the Gray map the obtained Z4-linear codes have the same parameters and fundamental properties as the codes in the usual binary linear Reed-Muller family. To make more evident the duality relationships in the constructed families the concept of Kronecker inner product is introduced.
In this paper, two different Gray-like maps from $Z_p^alphatimes Z_{p^k}^beta$, where $p$ is prime, to $Z_p^n$, $n={alpha+beta p^{k-1}}$, denoted by $phi$ and $Phi$, respectively, are presented. We have determined the connection between the weight en
umerators among the image codes under these two mappings. We show that if $C$ is a $Z_p Z_{p^k}$-additive code, and $C^bot$ is its dual, then the weight enumerators of the image $p$-ary codes $phi(C)$ and $Phi(C^bot)$ are formally dual. This is a partial generalization of [On $Z_{2^k}$-dual binary codes, arXiv:math/0509325], and the result is generalized to odd characteristic $p$ and mixed alphabet. Additionally, a construction of $1$-perfect additive codes in the mixed $Z_p Z_{p^2} ... Z_{p^k}$ alphabet is given.
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.
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 gener
alization. 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.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
