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تسجيل مستخدم جديدNew completely regular q-ary codes based on Kronecker products
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For any integer $rho geq 1$ and for any prime power q, the explicit construction of a infinite family of completely regular (and completely transitive) q-ary codes with d=3 and with covering radius $rho$ is given. The intersection array is also computed. Under the same conditions, the explicit construction of an infinite family of q-ary uniformly packed codes (in the wide sense) with covering radius $rho$, which are not completely regular, is also given. In both constructions the Kronecker product is the basic tool that has been used.
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This work is a survey on completely regular codes. Known properties, relations with other combinatorial structures and constructions are stated. The existence problem is also discussed and known results for some particular cases are established. In p
articular, we present a few new results on completely regular codes with covering radius 2 and on extended completely regular codes.
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Given a parity-check matrix $H_m$ of a $q$-ary Hamming code, we consider a partition of the columns into two subsets. Then, we consider the two codes that have these submatrices as parity-check matrices. We say that anyone of these two codes is the s
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ributions of a linear or additive code and its dual. In particular, for each case, we determine the dual scheme, on the same set but with different metric, such that the weight distribution of an additive code $C$ in the Doob scheme $D(m,n+n)$ is related by the MacWilliams identities with the weight distribution of the dual code $C^perp$ in the dual scheme. We note that in the case of a linear code $C$ in $E_4^m times F_4^{n}$, the weight distributions of $C$ and $C^perp$ in the same scheme are also connected.
Let $q=2^n$, $0leq kleq n-1$, $n/gcd(n,k)$ be odd and $k eq n/3, 2n/3$. In this paper the value distribution of following exponential sums [sumlimits_{xin bF_q}(-1)^{mathrm{Tr}_1^n(alpha x^{2^{2k}+1}+beta x^{2^k+1}+ga x)}quad(alpha,beta,gain bF_{q})]
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