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Register a new userThe $(n,m,k,lambda)$-Strong External Difference Family with $m geq 5$ Exists
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Added by
Jiejing Wen
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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The notion of strong external difference family (SEDF) in a finite abelian group $(G,+)$ is raised by M. B. Paterson and D. R. Stinson [5] in 2016 and motivated by its application in communication theory to construct $R$-optimal regular algebraic manipulation detection code. A series of $(n,m,k,lambda)$-SEDFs have been constructed in [5, 4, 2, 1] with $m=2$. In this note we present an example of (243, 11, 22, 20)-SEDF in finite field $mathbb{F}_q$ $(q=3^5=243).$ This is an answer for the following problem raised in [5] and continuously asked in [4, 2, 1]: if there exists an $(n,m,k,lambda)$-SEDF for $mgeq 5$.
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We find three families of twisting maps of K^m with K^n. One of them is related to truncated quiver algebras, the second one consists of deformations of the first and the third one requires m=n and yields algebras isomorphic to M_n(K). Using these families and some exceptional cases we construct all twisting maps of K^3 with K^3.
Strong external difference family (SEDF) and its generalizations GSEDF, BGSEDF in a finite abelian group $G$ are combinatorial designs raised by Paterson and Stinson [7] in 2016 and have applications in communication theory to construct optimal strong algebraic manipulation detection codes. In this paper we firstly present some general constructions of these combinatorial designs by using difference sets and partial difference sets in $G$. Then, as applications of the general constructions, we construct series of SEDF, GSEDF and BGSEDF in finite fields by using cyclotomic classes.
BCH codes are an interesting class of cyclic codes due to their efficient encoding and decoding algorithms. In many cases, BCH codes are the best linear codes. However, the dimension and minimum distance of BCH codes have been seldom solved. Until now, there have been few results on BCH codes over $gf(q)$ with length $q^m+1$, especially when $q$ is a prime power and $m$ is even. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. The BCH codes presented in this paper have good parameters in general, and contain many optimal linear codes.
In this paper, we study the $sigma$-self-orthogonality of constacyclic codes of length $p^s$ over the finite commutative chain ring $mathbb F_{p^m} + u mathbb F_{p^m}$, where $u^2=0$ and $sigma$ is a ring automorphism of $mathbb F_{p^m} + u mathbb F_{p^m}$. First, we obtain the structure of $sigma$-dual code of a $lambda$-constacyclic code of length $p^s$ over $mathbb F_{p^m} + u mathbb F_{p^m}$. Then, the necessary and sufficient conditions for a $lambda$-constacyclic code to be $sigma$-self-orthogonal are provided. In particular, we determine the $sigma$-self-dual constacyclic codes of length $p^s$ over $mathbb F_{p^m} + u mathbb F_{p^m}$. Finally, we extend the results to constacyclic codes of length $2 p^s$.
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