On the Covering Radius o f Codes 385
نویسنده
چکیده
The covering radius R of a code is the maximal distance of any vector from the code. This work gives a number of new results concerning t[ n, k], the minimal covering radius of any binary code of length n and dimension k. For example r[ n, 41 and t [ n, 51 are determined exactly, and reasonably tight bounds on t[ n, k] are obtained for any k when n is large. These results are found by using several new constructions for codes with small covering radius. One construction, the amalgamated direct sum, involves a quantity called the norm of a code. Codes with norm 5 2 R + 1 are called normal, and may be combined efficiently. The paper concludes with a table giving bounds on r [ n , k] for n I 64.
منابع مشابه
On the covering radius of some binary cyclic codes
We compute the covering radius of some families of binary cyclic codes. In particular, we compute the covering radius of cyclic codes with two zeros and minimum distance greater than 3. We compute the covering radius of some binary primitive BCH codes over F2f , where f = 7, 8.
متن کاملOn the covering radius of small codes versus dual distance
Tietäväinen’s upper and lower bounds assert that for block-length-n linear codes with dual distance d, the covering radius R is at most n2 − ( 2 − o(1)) √ dn and typically at least n2 − Θ( √ dn log nd ). The gap between those bounds on R − n2 is an Θ( √ log nd ) factor related to the gap between the worst covering radius given d and the sphere-covering bound. Our focus in this paper is on the c...
متن کاملFurther Results on the Covering Radii of the Reed-Muller Codes
Let R(r, m) by the rth order Reed-Muller code of length 2 )n, and let o(r, m) be its covering radius. We obtain the following new results on the covering radius of R(r, m): 1. p(r + 1, m + 2) > 2p(r, m) + 2 if 0 < r < m 2. This improves tile successive use of the known inequalities p(r + 1, m + 2) _> 20 (r + 1, m + 1) and p (r + 1, m + 1) -> O (r, m). 2. P (2, 7) -< 44. Previously best known up...
متن کاملInternational Journal of Mathematics And its Applications On Covering Radius of Codes Over R = Z 2 + u Z 2 , where u 2 = 0 Using Bachoc Distance
In this paper, we give lower and upper bounds on the covering radius of codes over the ring R = Z2 + uZ2, where u2 = 0 with bachoc distance and also obtain the covering radius of various Repetition codes, Simplex codes of α-Type code and β-Type code. We give bounds on the covering radius for MacDonald codes of both types over R = Z2 + uZ2. MSC: 20C05, 20C07, 94A05, 94A24.
متن کاملA note on the covering ra optimum codes
Bhandari, M.C. and M.S. Garg, A note on the covering radius of optimum codes, Discrete Applied Mathematics 33 (1991) 3-9. This paper gives a lower bound and an upper bound for the covering radius of optimum codes. The upper bound so obtained is better than other known upper bounds, restricted to optimum codes. Optimum codes of covering radius d1 and d2 are shown to be normal. A binary linear co...
متن کامل