Affiliation:
1. Department of Mathematics , University of Aveiro , 3810-193 Aveiro , Portugal
Abstract
Abstract
The Golomb-Welch conjecture states that there is no perfect r-error correcting Lee code of word length n over ℤ for n ≥ 3 and r ≥ 2. This problem has received great attention due to its importance in applications in several areas beyond mathematics and computer sciences. Many results on this subject have been achieved, however the conjecture is only solved for some particular values of n and r, namely: 3 ≤ n ≤ 5 and r ≥ 2; n = 6 and r = 2. Here we give an important contribution for the case n = 7 and r = 2, establishing cardinality restrictions on codeword sets.
Reference16 articles.
1. Golomb S.W., Welch L.R., Perfect codes in the Lee metric and the packing of polyominoes, SIAM J. Appl. Math., 1970, 18, 302-317.
2. Lee C.Y., Some properties of nonbinary error-correcting codes, IRE Trans. Inf. Theory, 1958, 4, 72-82.
3. Golomb S.W., Welch L.R., Algebraic coding and the Lee metric, In: Error Correcting Codes, Wiley, New York, 1968, 175-189.
4. Barg A., Mazumdar A., Codes in permutations and error correction for rank modulation, IEEE Trans. Inf. Theory, 2010, 56(7), 3158-3165.
5. Blaum M., Bruck J., Vardy A., Interleaving schemes for multidimensional cluster errors, IEEE Trans. Inf. Theory, 1998, 44, 730-743.