A. V. Kholin, âShortest path computing circuit,â Soviet patent appl. SU. 552 617, 1977. A. A. Lelis, V. A. Klishin, and G. S. Polishchuk, âTopological graph shortest ...
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 39, NO. 4, JULY 1993
[71 J.L. Massey “Error bounds for tree codes, trellis codes, and convolutional codes with encoding and decoding procedures,”in Coding and
Complexity, G. Longo, Ed. New York: Springer, 1976.
PI A. J. Viterbi and J. K. Omura, Principles of Digital Communication and Coding.
New York: McGraw-Hill, 1979.
[91 A. J. Viterbi, “Error bounds for convolutional codes and an asymptotically optimum decoding algorithm,” IEEE Trans. Inform. Theory, vol. IT-13, Apr. 1967, pp. 26&269. WI R. Bellman and S. Dreyfus, Applied Dynamic Programming. Princeton, NJ: Princeton Univ. Press, 1962. [ill G. J. Minty, “A comment on the shortest route problem,” Oper. Rex, vol. 5, p. 724, Oct. 1957. WI R. S. Muller and T. I. Kamins, Device Ekctronics for Integrated Circuits. New York: Wiley, 1977. [I31 J.B. Dennis, Mathematical Programming and Electrical Networks. New York: Wiley, 1959. [I41 M. Iri, Network Flow, Transportation and Scheduling-Theory and Algorithms. New York: Academic Press, 1969. WI G. V. Karandakov,“Network shortestpath detector-Is simplified using spark elektrodesbetween graph nodes,”Soviet patent appl. SU 397 931, 1974. P61 L. V. Fedotov, V. 1. Mikhailenk, and S. V. Ozirskii, “Graph shortest path determination apparatus,”Soviet patent appl. SU 1 488 824, 1989. 1171 A. V. Kholin, “Shortest path computing circuit,” Soviet patent appl. SU 552 617, 1977. 1181A. A. Lelis, V. A. Klishin, and G. S. Polishchuk, “Topological graph shortestpath finder,” Soviet patent appl. SU 1 314 354, 1987.
Normal and Abnormal
Codes
Tuvi Etzion, Member, IEEE, Gadi Greenberg, and Iiro S. Honkala
1453
CP={cEC:
c; = u} then C is normal if there exists a coordinate
i such that d(z, C$‘) + d(z, C,‘“‘) 5 2R + 1,
for z E F,“,
(1)
where d(z, 4) = n. If (1) holds for coordinate i we say that coordinate i is acceptable. If C is not normal then it is an abnormal code. An interesting question in this context is to determine which codes are normal and which codes are abnormal. One important factor is the ratio between the covering radius of the code and its minimum distance. van Wee [8] proved that all (n, 2R)R codes and all (n, 2R + 1)R codes are normal. Hou [5] proved that all linear quasi-perfect codes (codes with covering radius R and minimum distance at least 2R - l), are normal. In Section II we prove that an (n, 2R - 1)R code is normal, if R does not divide n. Frankl [6] proved that there are abnormal codes with covering radius 1. van Wee [8] (see also Honkala and Hamlllinen [3]) showed that for each R there exists no such that for each n 2 no there exists an abnormal code with covering radius R. It is not known whether linear abnormal codes exist. In Section III we give a construction for abnormal (n, R)R codes if some conditions hold. We apply this construction to obtain such codes for R 5 6. Another construction produce an abnormal (n, d - 1)R + 1 code if there exists an (n, d)R code and some conditions hold. Consequences of this construction are that .for each R 2 1, there exists an no, such that for each n 2 no there exists an abnormal (n, R - l)R code, and for each R 2 1 there exists an ma such that for all m > mo there exists either an abnormal (2m - 1, R)R code or an abnormal (2m, R)R code. Another consequence is the existence of abnormal (n, 3)2, (n, 4)3, and (n, 5)4 codes. II. NORMAL QUASI-PERFECT CODES
Abstract-It is proved that codes of length n, covering radius Z?, and minimum Hamming distance 2R-1 are normal if R does not divide n. Constructions for abnormal codes with covering radius R and minimum Hamming distance at least R-l are given. Index Terms- BCH code, covering radius, Hamming code, Preparata code, quasi-perfect code.
code, normal
I. INTRODUCTION
Much research in the area of covering radius is on the normality of the codes. The main reason is that by using the amalgamated direct sum [l] one can generate from normal codes sparse covering codes with larger covering radius. For two words z = zlza . e . x,, y = yrya . . . yn of length n the Humming distance (distance in short) between z and y, d(z,y) is defined by d(s,y) = ]{i: zi # y;}], the support of 2, supp (z) = {i:~, = l}, and the weight of Z, W(Z) = [ supp (a~)]. The minimum distance of a code C, d(C) = minI,YEc d(z,y). For a word 2, d(z, C) = min,cc d(~, c) is the d%nce of x from C. The covering radius of C is R if R = rnax,EF; d(z, C). An (n, d)R code C is a code of length n, covering radius at most R, and minimum distance at least d. If Manuscript received March 26, 1992. T. Etzion is with the Computer ScienceDepartment,Technion, Haifa 32000, Israel. This work was supportedin part by Technion V.P.R. Fund and in part by the fund for the promotion of researchat the Technion. G. Greenbergis with the Mathematics Department,Technion, Haifa 32000, Israel. I. S. Honkala is with the Department of Mathematics, University of Turku, 20500 Turku, Finland. IEEE Log Number 9209054.
As previously stated it is known that all linear quasi-perfect codes are normal [5] and (n, d)R codes with d 2 2R are normal [8]. These results are strengthened with the following theorem. Theorem 1: If C is an (n, 2R - l)R code, where R does not divide n, then C is normal and all its coordinates are acceptable. Proof: Assume the contrary, i.e., that one of the coordinated is not acceptable. W.1.o.g. we can assume it is the first coordinate. Let x E Fc be a word for which d(z 1C(l)) 1 0 + d(z ) C(l))
> 2R + 1.
(2)
Let t = d(z, C) and let c E C be a codeword for which d(z, c) = t. W.1.o.g. we can assume that c is the all zero codeword and hence t = W(x). We distinguish between two cases. Case 1: 0 5 t 5 R - 1. Again w.1.o.g. we can assume that the t l’s of z are in the first R coordinates. Consider the word yo = 1
R+lOn--R-l
Let zo E C be the codeword for which d(yo, 20) 5 R. Since W (ya) = R + 1 and d(C) = 2R - 1, it follows that 2R - 1 2 W(z0) 5 2R + 1. If W(z0) 2 2R then za covers all the l’s of ya and hence, a0 E Cir). Therefore, we have d(z, C,$“, + d(s, CT,“‘) < d(z, 0) + d(?, zo)
