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1.
Jürgen Bierbrauer 《Designs, Codes and Cryptography》2009,50(2):235-241
It has been proved in Bierbrauer and Kyureghyan (Des. Codes Cryptogr. 46:269–301, 2008) that a binomial function aX
i
+ bX
j
can be crooked only if both exponents i, j have 2-weight ≤2. In the present paper we give a brief construction for all known examples of crooked binomial functions.
These consist of an infinite family and one sporadic example. The construction of the sporadic example uses the properties
of an algebraic curve of genus 3. Computer experiments support the conjecture that each crooked binomial is equivalent either
to a member of the family or to the sporadic example.
相似文献
2.
Anuradha Sharma Gurmeet K. Bakshi Madhu Raka 《Finite Fields and Their Applications》2007,13(4):1086-1095
Let m be a positive integer and q be an odd prime power. In this paper, the weight distributions of all the irreducible cyclic codes of length 2m over Fq are determined explicitly. 相似文献
4.
Anuradha Sharma Gurmeet K. Bakshi Madhu Raka 《Finite Fields and Their Applications》2007,13(4):1071-1085
Polyadic codes constitute a special class of cyclic codes and are generalizations of quadratic residue codes, duadic codes, triadic codes, m-adic residue codes and split group codes, which have good error-correcting properties. In this paper, we give necessary and sufficient conditions for the existence of polyadic codes of prime power length. Examples of some good codes arising from the family of polyadic codes of prime power length are also given. 相似文献
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Quantum error-correcting codes with good parameters can be constructed by evaluating polynomials at the roots of the polynomial trace [18]. In this paper, we propose to evaluate polynomials at the roots of trace-depending polynomials (given by a constant plus the trace of a polynomial) and show that this procedure gives rise to stabilizer quantum error-correcting codes with a wider range of lengths than in [18] and with excellent parameters. Namely, we are able to provide new binary records according to [21] and non-binary codes improving the ones available in the literature. 相似文献
7.
Anuradha Sharma Gurmeet K. Bakshi V. C. Dumir Madhu Raka 《Finite Fields and Their Applications》2004,10(4):133
Let q be an odd prime power and p be an odd prime with gcd(p,q)=1. Let order of q modulo p be f,
and qf=1+pλ. Here expressions for all the primitive idempotents in the ring Rpn=GF(q)[x]/(xpn−1), for any positive integer n, are obtained in terms of cyclotomic numbers, provided p does not divide λ if n2. The dimension, generating polynomials and minimum distances of minimal cyclic codes of length pn over GF(q) are also discussed. 相似文献
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9.
Crooked functions are combinatorial objects of great interest. It is already known that the only monomial and binomial crooked functions are quadratic. In this paper, we investigate conditions on the shape of a polynomial to be crooked. Furthermore, the notion of exceptional crooked is introduced, similarly to those of APN or PN exceptional functions. Via a connection with algebraic varieties over finite fields, we provide non-existence results of exceptional crooked functions. 相似文献
10.
A perfect (v,{ki∣1≤i≤s},ρ) difference system of sets (DSS) is a collection of s disjoint ki-subsets Di, 1≤i≤s, of any finite abelian group G of order v such that every non-identity element of G appears exactly ρ times in the multiset {a−b∣a∈Di,b∈Dj,1≤i≠j≤s}. In this paper, we give a necessary and sufficient condition in terms of Jacobi sums for a collection {Di∣1≤i≤s} defined in a finite field Fq of order q=ef+1 to be a perfect (q,{ki∣1≤i≤s},ρ)-DSS, where each Di is a union of cyclotomic cosets of index e (and the zero 0∈Fq). Also, we give numerical results for the cases e=2,3, and 4. 相似文献
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Let be the finite field with q elements, and T a positive integer. In this article, we find an asymptotic formula for the total number of monic irreducible binomials in of degree less or equal to T, when T is large enough. We also show explicit lower and upper bounds for the number of binomials in the case when T is small. 相似文献
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New families of good q-ary (q is an odd prime power) Calderbank-Shor-Steane (CSS) quantum codes derived from two distinct classical Bose-Chaudhuri-Hocquenghem (BCH) codes, not necessarily self-orthogonal, are constructed. These new families consist of CSS codes whose parameters are better than the ones available in the literature and comparable to the parameters of quantum BCH codes generated by applying the q-ary Steane’s enlargement of CSS codes. 相似文献
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18.
MDS self-dual codes over finite fields have attracted a lot of attention in recent years by their theoretical interests in coding theory and applications in cryptography and combinatorics. In this paper we present a series of MDS self-dual codes with new length by using generalized Reed-Solomon codes and extended generalized Reed-Solomon codes as the candidates of MDS codes and taking their evaluation sets as a union of cyclotomic classes. The conditions on such MDS codes being self-dual are expressed in terms of cyclotomic numbers. 相似文献
19.
Jürgen Bierbrauer 《Designs, Codes and Cryptography》2002,25(2):189-206
We present a new approach to the theory of cyclic and constacyclic codes and generalize the theory to cover the family of additive (not necessarily linear) cyclic codes. The approach is based on the action of the Galois group (cyclotomic cosets). The conventional representation of cyclic codes as ideals in a factor ring of the polynomial ring is not needed. 相似文献
20.
Manju Pruthi 《Proceedings Mathematical Sciences》2001,111(4):371-379
In this paper explicit expressions ofm + 1 idempotents in the ring
are given. Cyclic codes of length 2
m
over the finite fieldF
q, of odd characteristic, are defined in terms of their generator polynomials. The exact minimum distance and the dimension
of the codes are obtained. 相似文献