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1.
In this paper, twenty new codes of dimension 6 are presented which give improved bounds on the maximum possible minimum distance of quaternary linear codes. These codes belong to the class of quasi-twisted (QT) codes, and have been constructed using a stochastic optimization algorithm, tabu search. A table of upper and lower bounds for d 4(n,6) is presented for n≤ 200. Received: 20 December 1996 / Accepted: 13 May 1997  相似文献   

2.
A code of lengthn, dimensionk and minimum distanced ismaximum distance separable (MDS) ifk+d=n+1. We give the number of MDS codes of length 7 and dimension 3 on finite fields withq elements whereq=2 m . In order to get this number, we compute the number of configurations of seven points in the projective plane overF q , no three of which are collinear.  相似文献   

3.
 The weight hierarchy of a linear [n, k; q] code C over GF(q) is the sequence (d 1, d 2, . . . , d k ) where d r is the smallest support of an r-dimensional subcode of C. An [n, k; q] code is external non-chain if for any r and s, where 1≦r<sk, there are no subspaces D and E, such that DE, dim D=r, dim E=s, w S (D)=d r , and w S (E)=d s . Bounds on the weight hierarchies of such codes of dimension 4 are studied. Received: September 27, 1996  相似文献   

4.
Let F n be the n-dimensional vector space over ℤ2. A (binary) 1-perfect partition of F n is a partition of F n into (binary) perfect single error-correcting codes or 1-perfect codes. We define two metric properties for 1-perfect partitions: uniformity and distance invariance. Then we prove the equivalence between these properties and algebraic properties of the code (the class containing the zero vector). In this way, we characterize 1-perfect partitions obtained using 1-perfect translation invariant and not translation invariant propelinear codes. The search for examples of 1-perfect uniform but not distance invariant partitions enabled us to deduce a non-Abelian propelinear group structure for any Hamming code of length greater than 7. Received: March 6, 2000; revised version: November 30, 2000  相似文献   

5.
Codes C 1 ,…,C M of length n over ? q and an M × N matrix A over ? q define a matrix-product code C = [C 1 C M ] ·A consisting of all matrix products [c 1 … c M ] ·A. This generalizes the (u|u+v)-, (u+v+w|2u+v|u)-, (a+x|b+x|a+b+x)-, (u+v|u-v)- etc. constructions. We study matrix-product codes using Linear Algebra. This provides a basis for a unified analysis of |C|, d(C), the minimum Hamming distance of C, and C . It also reveals an interesting connection with MDS codes. We determine |C| when A is non-singular. To underbound d(C), we need A to be `non-singular by columns (NSC)'. We investigate NSC matrices. We show that Generalized Reed-Muller codes are iterative NSC matrix-product codes, generalizing the construction of Reed-Muller codes, as are the ternary `Main Sequence codes'. We obtain a simpler proof of the minimum Hamming distance of such families of codes. If A is square and NSC, C can be described using C 1 , …,C M and a transformation of A. This yields d(C ). Finally we show that an NSC matrix-product code is a generalized concatenated code. Received: July 20, 1999; revised version: August 27, 2001  相似文献   

6.
A maximum distance separable (MDS) block code is a linear code whose distance is maximal among all linear block codes of rate k/n. It is well known that MDS block codes do exist if the field size is more than n. In this paper we generalize this concept to the class of convolutional codes of a fixed rate k/n and a fixed code degree δ. In order to achieve this result we will introduce a natural upper bound for the free distance generalizing the Singleton bound. The main result of the paper shows that this upper bound can be achieved in all cases if one allows sufficiently many field elements. Received: December 10, 1998; revised version: May 14, 1999  相似文献   

7.
Complete (n, k)-arcs in PG(k − 1, q) and projective (n, k) q -AMDS codes that admit no projective extensions are equivalent objects. We show that projective AMDS codes of reasonable length admit only linear extensions. Thus, we are able to prove the maximality of many known linear AMDS codes. At the same time our results sharply limit the possibilities for constructing long nonlinear AMDS codes. We also show that certain short linear AMDS codes are maximal. Central to our approach is the Bruen–Silverman model of linear codes first introduced in Alderson (On MDS codes and Bruen–Silverman codes. Ph.D. Thesis, University of Western Ontario, 2002) and Alderson et al. (J. Combin. Theory Ser. A 114(6), 1101–1117, 2007). The authors acknowledge support from the N.S.E.R.C. of Canada.  相似文献   

8.
In this note, a class of error-correcting codes is associated to a toric variety defined over a finite field q, analogous to the class of AG codes associated to a curve. For small q, many of these codes have parameters beating the Gilbert-Varshamov bound. In fact, using toric codes, we construct a (n,k,d)=(49,11,28) code over 8, which is better than any other known code listed in Brouwers tables for that n, k and q. We give upper and lower bounds on the minimum distance. We conclude with a discussion of some decoding methods. Many examples are given throughout.  相似文献   

9.
In this paper we obtain an effective Nullstellensatz using quantitative considerations of the classical duality theory in complete intersections. Letk be an infinite perfect field and let f1,...,f n–rk[X1,...,Xn] be a regular sequence with d:=maxj deg fj. Denote byA the polynomial ringk [X1,..., Xr] and byB the factor ring k[X1,...,Xn]/(f1,...,fn r); assume that the canonical morphism AB is injective and integral and that the Jacobian determinant with respect to the variables Xr+1,...,Xn is not a zero divisor inB. Let finally B*:=HomA(B, A) be the generator of B* associated to the regular sequence.We show that for each polynomialf the inequality deg (¯f) dn r(+1) holds (¯fdenotes the class off inB and is an upper bound for (n–r)d and degf). For the usual trace associated to the (free) extensionA B we obtain a somewhat more precise bound: deg Tr(¯f) dn r degf. From these bounds and Bertini's theorem we deduce an elementary proof of the following effective Nullstellensatz: let f1,..., fs be polynomials in k[X1,...,Xn] with degrees bounded by a constant d2; then 1 (f1,..., fs) if and only if there exist polynomials p1,..., psk[X1,..., Xn] with degrees bounded by 4n(d+ 1)n such that 1=ipifi. in the particular cases when the characteristic of the base fieldk is zero ord=2 the sharper bound 4ndn is obtained.Partially supported by UBACYT and CONICET (Argentina)  相似文献   

10.
There are given k Poisson processes with parameters (rates of occurrence) λ1, …, λ k . Let λ(1) ≤ λ(2) ≤ … ≤ λ(k) denote the ordered set of values of the parameters. A procedure is given for selecting the process corresponding to λ(k) and estimating its parameter (λ(k)). The given procedure controls the joint risk of improper selection and of large error in the estimate. Let θ > 1 and 0 < α, β < 1 be given numbers, and let δ denote the estimate of λ(k). The joint probability that a correct selection is made and that |(δ/λ(k)) ? 1| ≤ α is at least as large as β, for (λ(k)(k?1)) ≥ θ. Two cases are considered, that is, when the processes are observed continuously in time, and when they are observed at successive intervals of time. Both the cases lead to the same theoretical results.  相似文献   

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