COUNTING THE GENERATOR MATRICES OF Z2Z8-CODES

Year 2013, Volume: 1 Issue: 2, 143 - 149, 01.12.2013

İrfan Şiap İsmailaydoğdu

Abstract

In this paper, we count the number of matrices whose rows generate different Z2Z8additive codes. This is a natural generalization of thewell known Gaussian numbers that count the number of matrices whose rowsgenerate vector spaces with particular dimension over finite fields.Due tothis similarity we name this numbers as Mixed Generalized Gaussian Numbers (MGN). By specialization of MGN formula the well known formula forthe number of binary codes and the number of codes over Z8, and for additive Z2Z4codes are easily derived. Also, we conclude by some properties andexamples of the MGN numbers that provide a good source for new number sequences that are not listed in The On-Line Encyclopedia of Integer Sequences

Keywords

Gaussian numbers , Mixed Generalized Gaussian Numbers , NewSequences

References

• Aydogdu, I. and Siap, I., The Structure of ZZ2sAdditive Codes: Bounds on the minimum distance, Applied Mathematics & Information Sciences (AMIS), 7, 6, 2271-2278 (2013).
• Bilal, M., Borges, J., Dougherty, S., Fernandez, C., Optimal Codes over Z2Z4In libro de acts VII Jornadas de Matematica Discreta i Algoritmica, Castro Urdiales (Spain), 131-139, (2010).
• Bilal, M., Borges, J., Dougherty, S., Fernandez, C., Extensions of Z2Z4-additive self-dual codes preserving their properties, IEEE International Symposium on Information Theory , 3101- 3105 (2012).
• Bona, M., Combinatorics of permutations,Discrete Mathematics and Its Applications, Chap- man and Hall/CRC, (2004).
• Borges, J., Fernandez, C., Pujol, J., Rifa, J. and Villanueva, M., On Z2Z4-linear codes and duality, V Jornadas de Matematica Discreta i Algoritmica, Soria (Spain), Jul. 11-14, 171-177, (2006).
• Borges, J., Fernandez-Cordoba, C., Pujol, J., Rifa, J. and Villanueva, M., ZZ-linear codes: generator matrices and duality, Designs, Codes and Cryptography, 54 (2), 167-179, (2010).
• Brouwer, A.E., Hamalainen,H.O., Ostergard, P.R.J., Sloane, N.J.A., Bounds on Mixed Bi- nary/Ternary Codes, IEEE Transactions on Information Theory 44 (1): 140-161 (1998)
• Delsarte, P., An algebraic approach to the association schemes of coding theory, Philips Re- search Rep.Supp., 10, vi+97 (1973). [9] Delsarte, P., Levenshtein, Trans.Inform.Theory, 44 (6) 2477-2504 (1998). V.:Association schemes and coding theory, IEEE
• Hammons, A.R., Kumar, V., Calderbank, A.R., Sloane, N.J.A., Sol´e, P., The Z4-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 301-319 (1994).
• MacWilliams, F.J., and Sloane, N.J.A., The Theory of Error-Correcting Codes, North- Holland: New York, NY, (1977).
• Pujol J., Rifa J., Translation invariant propelinear codes, IEEE Trans. Inform. Theory 43 590-598 (1997).
• Salturk E., Siap I., On Generalized Gaussian Numbers, Albanian Journal of Mathematics, 6, 2 87-102 (2012).
• Salturk E., Siap I., Generalized Gaussian Numbers Related to Linear Codes over Galois Rings,European Journal of Pure and Applied Mathematics 5 250-259 (2012).
• Salturk E., Siap I., Generalized Gaussian Numbers and Some New Sequences, Physica Mace- donica, accepted 6, (2013).
• Sloane, N., The On-Line Encyclopedia of Integer Sequences (OEIS), (”http://oeis.org/” ac- cessed on March 8th, 2013). Yildiz Technical University, Faculty of Arts and Science, Department of Mathe
• matics, Istanbul-Turkey E-mail address: [e-mail],[e-mail]