Linear programming bounds for codes of small size

2002

Abstract Let A (n, d) denote the maximum possible number of codewords in a binary code of length n and minimum Hamming distance d. For large values of n, the best known upper bound, for fixed d, is the Johnson bound. We give a new upper bound which is at least as good as the Johnson bound for all values of n and d, and for each d there are infinitely many values of n for which the new bound is better than the Johnson bound.

Ieee Transactions on Information Theory, 1998

k01 j=0 dd=q j e = N d + A + k 0 I d 0 2; so k n 0 N d + I d 0 A + 2: Comparing this inequality with Theorem 4, we completed the proof. In the following, we give two examples. Example 1: q = 2, d = 5. If n > 12, the new bound is tighter than the Griesmer bound. For instance, the Griesmer bound cannot prove the nonexistence [13; 6; 5], but the new bound can. Example 2: q = 3, d = 9. If n > 13, the new bound is tighter than the Griesmer bound. For instance, the Griesmer bound cannot prove the nonexistence [14; 4; 9], but the new bound can. IV. CONCLUSION In this correspondence, we gave some bounds for [n; k; d] codes with the method of coset partitions. These bounds are new in type and are derived from the basic properties of the parity-check matrix. Comparing these bounds with tables of the best codes known, it can be shown that these bounds are not strong for arbitrary n and d although, indeed, Theorem 4 improves on the Griesmer bound for a large range of code parameters. Comparing with [3, Table I] in the binary case, we list some results in the following. Case 1: d = 3. In this case, Theorem 1 shows that k bn 0 log 2 (n + 1)c: The equality holds for Hamming [n = 2 r 0 1; k = n 0 r; d = 3] codes and also for shortened Hamming [n = 2 r 010m; k = n 0r 0m; d = 3] codes, where 0 < m < n 0 r. This shows that Theorem 1 is tight for d = 3. This also shows that the Hamming codes and shortened Hamming codes are optimal linear block codes for a given codelength and minimum distance d = 3, a fact known earlier in coding theory. Case 2: d = 5. We compare Theorem 2 with [3, Table I ]. In this case, Theorem 2 shows that k bn 0 4 0 log 2 (n 0 4)c. The results obtained are listed in Table I. Case 3: d = 7. In this case, Theorem 3 can be written as k bn 0 7 0 log 2 (n 0 7)c: The results obtained are listed in Table II. Case 4: d = 9. In this case, Theorem 4 can be written as k bn 0 12 0 log 2 (n 0 13)c:

IEEE Transactions on Information Theory, 2000

IEEE Transactions on Information Theory, 1989

MEMBER, IEEE with M (which is expected). However, as M increases beyond Ahstrart-Some new lower bounds are given for A(n,4, IV), the maximum number of codewords in a binary code of length n, minimum distance 4, and constant weight IV. In a number of cases the results significantly 1.0 improve on the best bounds previously known. h=O h=O 1. I. 141 S. S. Rappaport, "Demand assigned multiple access systems using collision type request channels: traffic capacity comparisons," IEEE Truns An (n, d, w) code is a code of length n, constant weight w, and

Designs, Codes and Cryptography, 1995

Motivated by applications in universal data compression algorithms we study the problem of bounds on the sizes of constant weight covering codes. We are concerned with the minimal sizes of codes of length n and constant weight u such that every word of length n and weight v is within Hamming distance d from a codeword. In addition to a brief summary of part of the relevant literature, we also give new results on upper and lower bounds to these sizes. We pay particular attention to the asymptotic covering density of these codes. We include tables of the bounds on the sizes of these codes both for small values ofn and for the asymptotic behavior. A comparison with techniques for attaining bounds for packing codes is also given. Some new combinatorial questions are also arising from the techniques.

IEEE Transactions on Information Theory, 2016

Multiply constant-weight codes (MCWCs) were introduced recently to improve the reliability of certain physically unclonable function response. In this paper, the bounds of MCWCs and the constructions of optimal MCWCs are studied. Firstly, we derive three different types of upper bounds which improve the Johnson-type bounds given by Chee et al. in some parameters. The asymptotic lower bound of MCWCs is also examined. Then we obtain the asymptotic existence of two classes of optimal MCWCs, which shows that the Johnson-type bounds for MCWCs with distances 2 m i=1 wi − 2 or 2mw − w are asymptotically exact. Finally, we construct a class of optimal MCWCs with total weight four and distance six by establishing the connection between such MCWCs and a new kind of combinatorial structures. As a consequence, the maximum sizes of MCWCs with total weight less than or equal to four are determined almost completely.

1998

We consider upper bounds on two fundamental parameters of a code; minimum distance and covering radius. New upper bounds on the covering radius of non-binary linear codes are derived by generalizing a method due to S. Litsyn and A. Tietäväinen lt: newu and combining it with a new upper bound on the asymptotic information rate of non-binary codes.

2007 IEEE International Symposium on Information Theory, 2007

Let Aq(n, d) be the maximum order (maximum number of codewords) of a q-ary code of length n and Hamming distance at least d. And let A(n, d, w) that of a binary code of constant weight w. Building on results from algebraic graph theory and Erdős-ko-Rado like theorems in extremal combinatorics, we show how several known bounds on Aq(n, d) and A(n, d, w) can be easily obtained in a single framework. For instance, both the Hamming and Singleton bounds can derived as an application of a property relating the clique number and the independence number of vertex transitive graphs. Using the same techniques, we also derive some new bounds and present some additional applications.

IEEE Transactions on Information Theory, 2005

We treat the problem of bounding components of the possible distance distributions of codes given the knowledge of their size and possibly minimum distance. Using the Beckner inequality from harmonic analysis, we derive upper bounds on distance distribution components which are sometimes better than earlier ones due to Ashikhmin, Barg, and Litsyn. We use an alternative approach to derive upper bounds on distance distributions in linear codes. As an application of the suggested estimates we get an upper bound on the undetected error probability for an arbitrary code of given size. We also use the new bounds to derive better upper estimates on the covering radius, as well as a lower bound on the error-probability threshold, as a function of the code's size and minimum distance.

IEEE Transactions on Information Theory, 1973

This paper presents a table of upper and lower bounds on rl,,,(rr,k), the maximum minimum distance over all binary, linear (n,k) error-correcting codes. The table is obtained by combining the best of the existing bounds on d,,,(n,k) with the mini&n distances of known codes and a variety of code-construction techniques.