1,533 research outputs found

    Linear Codes from Some 2-Designs

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    A classical method of constructing a linear code over \gf(q) with a tt-design is to use the incidence matrix of the tt-design as a generator matrix over \gf(q) of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of 22-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of 22-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterisation of highly nonlinear Boolean functions is presented

    Hadamard partitioned difference families and their descendants

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    If DD is a (4u2,2u2−u,u2−u)(4u^2,2u^2-u,u^2-u) Hadamard difference set (HDS) in GG, then {G,G∖D}\{G,G\setminus D\} is clearly a (4u2,[2u2−u,2u2+u],2u2)(4u^2,[2u^2-u,2u^2+u],2u^2) partitioned difference family (PDF). Any (v,K,λ)(v,K,\lambda)-PDF will be said of Hadamard-type if v=2λv=2\lambda as the one above. We present a doubling construction which, starting from any such PDF, leads to an infinite class of PDFs. As a special consequence, we get a PDF in a group of order 4u2(2n+1)4u^2(2n+1) and three block-sizes 4u2−2u4u^2-2u, 4u24u^2 and 4u2+2u4u^2+2u, whenever we have a (4u2,2u2−u,u2−u)(4u^2,2u^2-u,u^2-u)-HDS and the maximal prime power divisors of 2n+12n+1 are all greater than 4u2+2u4u^2+2u

    Cyclotomic Constructions of Skew Hadamard Difference Sets

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    We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields using unions of cyclotomic classes of order N=2p1mN=2p_1^m, where p1p_1 is a prime and mm a positive integer. Our main tools are index 2 Gauss sums, instead of cyclotomic numbers.Comment: 15 pages; corrected a few typos; to appear in J. Combin. Theory (A
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