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1.
The generalized Hamming weights of a linear code have been extensively studied since Wei first use them to characterize the cryptography performance of a linear code over the wire-tap channel of type II. In this paper, we investigate the generalized Hamming weights of three classes of linear codes constructed through defining sets and determine them partly for some cases. Particularly, in the semiprimitive case we solve a problem left in Yang et al. (2015) [30]. 相似文献
3.
The objectives of this paper are to survey and extend earlier results on the weight distributions of irreducible cyclic codes, present a divisibility theorem and develop bounds on the weights in irreducible cyclic codes. 相似文献
4.
Relative generalized Hamming weights (RGHWs) of a linear code with respect to a linear subcode determine the security of the linear ramp secret sharing scheme based on the linear codes. They can be used to express the information leakage of the secret when some keepers of shares are corrupted. Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems. In this paper, we investigate the RGHWs of cyclic codes of two nonzeros with respect to its irreducible cyclic subcodes. We give two formulae for RGHWs of the cyclic codes. As applications of the formulae, explicit examples are computed. Moreover, RGHWs of cyclic codes in the examples are very large, comparing with the generalized Plotkin bound of RGHWs. So it guarantees very high security for the secret sharing scheme based on the dual codes. 相似文献
6.
Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. In this paper, several classes of two-weight and three-weight linear codes are presented and their weight distributions are determined using Weil sums. Some of the linear codes obtained are optimal or almost optimal with respect to the Griesmer bound. 相似文献
7.
In this paper, a class of -ary linear codes with two weights is constructed by using the properties of cyclotomic classes of . The complete weight enumerators of these linear codes are also determined. In some cases, they are optimal and can be employed to obtain secret sharing schemes with interesting access structures and asymptotically optimal systematic authentication codes. 相似文献
9.
We construct a class of linear codes by choosing a proper defining set and determine their complete weight enumerators and weight enumerators. The results show that they have at most three weights and they are suitable for applications in secret sharing schemes. This is an extension of the results raised by Wang et al. (2017). 相似文献
10.
Journal of Algebraic Combinatorics - By appropriately choosing a defining set, we define a class of linear codes and establish their complete weight enumerators and weight enumerators using Weil... 相似文献
13.
The support of an [ n, k] linear code C over a finite field Fq is the set of all coordinate positions such that at least one codeword has a nonzero entry in each of these coordinate position. The rth generalized Hamming weight dr( C), 1 ⩽ r ⩽ k, of C is defined as the minimum of the cardinalities of the supports of all [ n, r] subcodes of C. The sequence ( d1( C), d2( C), … , dk( C)) is called the Hamming weight hierarchy (HWH) of C. The HWH, dr( C) = n − k + r; r = 1, 2 , …, k, characterizes maximum distance separable (MDS) codes. Therefore the matrix characterization of MDS codes is also the characterization of codes with the HWH dr( C) = n − k + r; r = 1, 2, … , k. A linear code C with systematic check matrix [ I∣ P], where I is the ( n − k) × ( n − k) identity matrix and P is a ( n − k) × k matrix, is MDS iff every square submatrix of P is nonsingular. In this paper we extend this characterization to linear codes with arbitrary HWH. Using this result, we characterize Near-MDS codes, Near-Near-MDS ( N2-MDS) codes and Aμ-MDS codes. The MDS-rank of C is the smallest integer η such that dη+1 = n − k + η + 1 and the defect vector of C with MDS-rank η is defined as the ordered set { μ1( C), μ2( C), μ3( C), … , μη( C), μη+1( C)}, where μi( C) = n − k + i − di( C). We call C a dually defective code if the defect vector of the code and its dual are the same. We also discuss matrix characterization of dually defective codes. Further, the codes meeting the generalized Greismer bound are characterized in terms of their generator matrix. The HWH of dually defective codes meeting the generalized Greismer bound are also reported. 相似文献
14.
Recently, linear codes with a few weights have been constructed and extensively studied due to their applications in secret sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, we construct several classes of linear codes with a few weights over , where is an odd prime. The weight distributions of these constructed codes are also settled by applications of the theory of quadratic forms and Gauss sums over finite fields. Some of the linear codes obtained are optimal or almost optimal. The parameters of these linear codes are new in most cases. Moreover, two classes of MDS codes are obtained. 相似文献
15.
We prove that given a binary Hamming code \({{\mathcal{H}}^n}\) of length n = 2 m ? 1, m ≥ 3, or equivalently a projective geometry PG( m ? 1, 2), there exist permutations \({\pi \in \mathcal{S}_n}\) , such that \({{\mathcal{H}}^n}\) and \({\pi({\mathcal{H}}^n)}\) do not have any Hamming subcode with the same support, or equivalently the corresponding projective geometries do not have any common flat. The introduced permutations are called AF permutations. We study some properties of these permutations and their relation with the well known APN functions. 相似文献
16.
In this paper we prove that a set of points (in a projective space over a finite field of q elements), which is incident with 0 mod r points of every hyperplane, has at least ( r−1) q+( p−1) r points, where 1< r< q= ph, p prime. An immediate corollary of this theorem is that a linear code whose weights and length have a common divisor r< q and whose dual minimum distance is at least 3, has length at least ( r−1) q+( p−1) r. The theorem, which is sharp in some cases, is a strong generalisation of an earlier result on the non-existence of maximal arcs in projective planes; the proof involves polynomials over finite fields, and is a streamlined and more transparent version of the earlier one. 相似文献
17.
Let m be a positive integer. Fix a nontrivial additive character for each finite field Fq. To state the first result of this paper, we also fix r distinct multiplicative characters 1,..., r for each finite field Fq with more than r elements. We shall prove that when varies over multiplicative characters of Fq other than the m-th roots of the r-tuples
of angles of Gauss sums are asymptotically equidistributed on the r-dimensional torus ( S1) r as q goes to infinity.The n-dimensional Kloosterman sum over Fq at a Fq× is
One can define the angle ( q, a) of Kln( q, a) in a suitable way. We shall prove that when a varies over nonzero elements of Fq, the q–1 angles ( q, am) of Kloosterman sums are asymptotically equidistributed as q goes to infinity. Mathematics Subject Classification (2000) 11L05, 14F20 相似文献
18.
Boolean functions have very nice applications in coding theory and cryptography. In coding theory, Boolean functions have been used to construct linear codes in different ways. The objective of this paper is to construct binary linear codes with few weights using the defining-set approach. The defining sets of the codes presented in this paper are defined by some special Boolean functions and some additional restrictions. First, two families of binary linear codes with at most three or four weights from Boolean functions with at most three Walsh transform values are constructed and the parameters of their duals are also determined. Then several classes of binary linear codes with explicit weight enumerators are produced. Some of the binary linear codes are optimal or almost optimal according to the tables of best codes known maintained at http://www.codetables.de, and the duals of some of them are distance-optimal with respect to the sphere packing bound. 相似文献
19.
Let \(\mathbb {F}_{q}\) be the finite field with \(q=p^{m}\) elements, where p is an odd prime and m is a positive integer. For a positive integer t, let \(D\subset \mathbb {F}^{t}_{q}\) and let \({\mathrm {Tr}}_{m}\) be the trace function from \(\mathbb {F}_{q}\) onto \(\mathbb {F}_{p}\). In this paper, let \(D=\{(x_{1},x_{2},\ldots ,x_{t}) \in \mathbb {F}_{q}^{t}\setminus \{(0,0,\ldots ,0)\} : {\mathrm {Tr}}_{m}(x_{1}+x_{2}+\cdots +x_{t})=0\},\) we define a p-ary linear code \(\mathcal {C}_{D}\) by $$\begin{aligned} \mathcal {C}_{D}=\{\mathbf {c}(a_{1},a_{2},\ldots ,a_{t}) : (a_{1},a_{2},\ldots ,a_{t})\in \mathbb {F}^{t}_{q}\}, \end{aligned}$$ where $$\begin{aligned} \mathbf {c}(a_{1},a_{2},\ldots ,a_{t})=({\mathrm {Tr}}_{m}(a_{1}x^{2}_{1}+a_{2}x^{2}_{2}+\cdots +a_{t}x^{2}_{t}))_{(x_{1},x_{2},\ldots ,x_{t}) \in D}. \end{aligned}$$ We shall present the complete weight enumerators of the linear codes \(\mathcal {C}_{D}\) and give several classes of linear codes with a few weights. This paper generalizes the results of Yang and Yao (Des Codes Cryptogr, 2016). 相似文献
20.
In this paper, the dual code of the binary cyclic code of length 2 n-1 with three zeros α, α t 1 and α t 2 is proven to have five nonzero Hamming weights in the case that n 4 is even and t1 = 2 n/2 + 1, t2 = 2 n-1-2 n/2+1 + 1 or 2 n/2 + 3, where α is a primitive element of the finite field F 2 n . The dual code is a divisible code of level n/2+1, and its weight distribution is also completely determined. When n = 4, the dual code satisfies Ward's bound. 相似文献
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