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1.
Binary block codes for correctingt symmetric, asymmetric and unidirectional errors are calledt-SyEC codes,t-AsEC codes andt-UEC codes respectively. Two tables with bounds on the cardinality of binary block codes for correcting asymmetric and unidirectional errors respectively are presented. They include many improvements over the existing literature. The lower bounds follow from explicit constructions, while the upper bounds are obtained by applying combinatorial arguments to the weight structure of such codes.The authors are with Department of Mathematics and Computing Science of Eindhoven University of Technology, The Netherlands. Part of this work was presented at the IEEE International Symposium on Information Theory, Budapest, 1991  相似文献   

2.
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   

3.
The atomic vectors of a finitely generated vector space C over a field F are characterized for C a subspace of the product vector space ? = ∏ i =1 n ? i over F. For finite fields, the minimal trellis diagram for mixed-codes is determined, and this provides the L-section minimal trellis diagram for linear codes. As an example, an extremely simple yet comprehensive analysis of the trellis structure of Reed-Muller codes is given. In particular, a trellis oriented generator matrix for the 2 l -section minimal trellis diagram of a Reed-Muller code is presented. Received: February 27, 1997; revised version: May 6, 1999  相似文献   

4.
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   

5.
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   

6.
Let [n, k, d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). Let d 5(n, k) be the maximum possible minimum Hamming distance of a linear [n, k, d; 5]-code for given values of n and k. In this paper, forty four new linear codes over GF(5) are constructed and a table of d 5(n, k) k≤ 8, n≤ 100 is presented.  相似文献   

7.
Error-correcting codes which are ideals in group rings where the underlying group is metacyclic and non-abelian are examined. Such a groupG(M, N,R) is the extension of a finite cyclic group M by a finite cyclic group N and has a presentation of the form (S, T:S M =1,T N =1, T· S=S R ·T) where gcd(M, R)=1, R N =1 modM, R 1. Group rings that are semi-simple, i.e., where the characteristic of the field does not divide the order of the group, are considered. In all cases, the field of the group ring is of characteristic 2, and the order ofG is odd.Algebraic analysis of the structure of the group ring yields a unique direct sum decomposition ofFG(M, N, R) to minimal two-sided ideals (central codes). In every case, such codes are found to be combinatorically equivalent to abelian codes and of minimum distance that is not particularly desirable. Certain minimal central codes decompose to a direct sum ofN minimal left ideals (left codes). This direct sum is not unique. A technique to vary the decomposition is described. p]Metacyclic codes that are one-sided ideals were found to display higher minimum distances than abelian codes of comparable length and dimension. In several cases, codes were found which have minimum distances equal to that of the best known linear block codes of the same length and dimension.  相似文献   

8.
Since the paper by Hammons e.a. [1], various authors have shown an enormous interest in linear codes over the ring ℤ4. A special weight function on ℤ4 was introduced and by means of the so called Gray map ϕ : ℤ4→ℤ2 2 a relation was established between linear codes over ℤ4 and certain interesting non-linear binary codes of even length. Here, we shall generalize these notions to codes over ℤ p2 where p is an arbitrary prime. To this end, a new weight function will be proposed for ℤ p2 . Further, properties of linear codes over ℤ p2 will be discussed and the mapping ϕ will be generalized to an isometry between ℤ p2 and ℤ p p , resp. between ℤ p2 n and ℤ p pn . Some properties of Galois rings over ℤ q will be described and two dual families of linear codes of length n = p m − 1, gcd(m, p) = 1, over ℤ q will be constructed. Taking q = p 2, their images under the new mapping can be viewed as a generalization of the binary Kerdock and the Preparata code, although they miss some of their nice combinatorial properties. Received: June 19, 2000; revised version: November 6, 2000  相似文献   

9.
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   

10.
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   

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