MACWILLIAMS IDENTITIES FOR POSET LEVEL WEIGHT ENUMERATORS OF LINEAR CODES

Codes over various metrics such as Rosenbloom-Tsfasman (RT), Lee, etc. have been considered in the literature. Recently, codes over poset metrics have been studied. Poset metric is a great generalization of many metrics especially the well-known ones such as the RT and the Hamming metrics. Poset metric can be realized on the channels with localized error occurrences. It has been shown that MacWilliams identities are not admissible for codes over poset metrics in general [15]. Lately, to overcome this problem some further studies on MacWilliams identities over poset metrics has been presented. In this paper, we introduce new poset level weight enumerators of linear codes over Frobenius commutative rings. We derive MacWilliams-type identities for each of the given enumerators which generalize in great deal the previous results discussed in the literature. Most of the weight enumerators in the literature such as Hamming, Rosenbloom-Tsfasman and complete m-spotty weight enumerators follow as corollaries to these identities especially.

Keywords:

MacWilliams identity, linear codes poset codes.,

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[1] Ahlswede R., Bassalygo L. A., and Pinsker M. S., Nonbinary codes correcting localized errors, IEEE Trans. Inform. Theory. 39 (1993) 1413-1416.
[2] Akbiyik S., Siap I., A P-complete weight enumerator with respect to poset metric and its MacWilliams identity (Turkish), Adiyaman University J. Sci. 1 (2011) 28-39.
[3] Akbiyik S., Siap I., MacWilliams Identities Over Some Special Posets, Communications Faculty of Sciences University of Ankara, Series A1: Mathematics and Statistics, Vol. 62 (1) (2013),61-71.
[4] Bassalygo L. A., Gelfand S. I., Pinsker M. S., Coding for channels with localized errors, Proc. 4th Soviet-Swedish Workshop in Inform. Theory, Sweden, 1989, pp. 95-99.
[5] Bassalygo L. A.,Gelfand S. I., Pinsker M. S., Coding for partially localized errors, IEEE Trans. Inform. Theory. 37 (1991) 880-884.
[6] Brualdi R. A., Graves J., Lawrence K. M., Codes with a poset metric, Discrete Math. 147 (1995) 57-72.
[7] Chang S. H., Rim M., Cosman P. C., Milstein L. B., Optimized unequal error protection using multiplexed hierarchical modulation, IEEE Trans. Inform. Theory. 58 (2012) 5816-5840.
[8] Claasen H. L., Goldbach R. W., A Field-like property of finite field, Indag. Math. 3 (1992) 11-26.
[9] Choi S., Hyun J. Y., Oh D. Y., Kim H. K., MacWilliams-type equivalence relations, available in arXiv:1205.1090v2.
[10] Felix L. V., Firer M., Canonical-systematic form for codes in hierarchical poşet metrics, Adv. Math. Commun. 6 (2012) 315-328.
[11] Firer M., Panek L., Rifo L., Coding in the presence of semantic value of information: Unequal error protection using poset decoders, available in arXiv:1108.3832v1 [cs.IT], Partial version in AIP Conf. Proc. 1490: 126-134.
[12] Fujiwara E., Code Design for Dependable Systems: Theory and Practical Applications, A John Wiley & Sons Inc. Pub., New Jersey, 2006.
[13] Gutierrez J. N., Tapia-Recillas H. , A MacWilliams Identity for Poset-Codes, Congr. Numer. 133 (1998) 63{73.
[14] Huang K., Liang C., Ma X., Bai B., Unequal error protection by partial superposition transmission using LDPC Codes. Available in arXiv:1309.3864 [cs.IT], 2013.
[15] Kim H. K., Oh D. Y., A Classi_cation of Posets Admitting the MacWilliams Identity, IEEE Transactions on Information Theory, 51 (4) (2005) 1424-1431.
[16] Korhonen J., Frossard P., Bit-error resilient packetization for streaming H.264/AVC video, Proc. of the 1st ACM International Workshop on Mobile Video, Germany, 2007, pp. 25-30.
[17] Kuriata E., Creation of unequal error protection codes for two groups of symbols, Int. J. Appl. Math. Comput. Sci. 18 (2008) 251-257.
[18] Larsson P., Codes for Correction of Localized Errors (Linkoping Studies in Science and Technology, dissertations, no. 374). Linkoping Sweden, (1995).
[19] MacWilliams F. J., Sloane N. J. A., The Theory of Error Correcting Codes, North-Holland Pub. Co., Amsterdam, 1977.
[20] Mardjuadi A., Weber J. H., Codes for multiple localized burst error correction, IEEE Trans. Inform. Theory. 44 (1998) 2020-2024.
[21] Masnik B., Wolf J., On linear unequal error protection codes, IEEE Trans. On Information Theory, IT- 13(4) (1967) 600607.
[22] Niederreiter H., Point sets and sequences with small discrepancy, Mh. Math. 104 (1987) 273-337.
[23] Niederreiter H., A combinatorial problem for vector spaces over finite fields, Discrete Math. 96 (1991) 221-228.
[24] Niederreiter H., Orthogonal arrays and other combinatorial aspects in the theory of uniform point distributions in unit cubes, Discrete Math. 106/107 (1992) 361-367.
[25] Ozen M., Siap V., The MacWilliams identity for m-spotty weight enumerators of linear codes over finitefi_elds, Comput. Math. Appl. 61 (2011) 1000-1004.
[26] Ozen M., Siap V., The MacWilliams identity for m-spotty Rosenbloom-Tsfasman weight enumerator, J. Franklin Inst. doi:10.1016/j.franklin.2012.06.002.
[27] Pinheiro J. A., Firer M., Classi_cation of Poset-Block Spaces Admitting MacWilliams-Type Identity, IEEE Trans. Inform. Theory. 58 12 (2012) 7246 - 7252.
[28] Roth R. M., Seroussi G., Location-correcting codes, IEEE Trans. Inform. Theory. 42 (1996) 554-565.
[29] Sharma A., Sharma A. K., MacWilliams type identities for some new m-spotty weight enumerators, IEEE Trans. Inform. Theory. 58 (2012) 3912-3924.
[30] Sharma A., Sharma A. K., On some new m-spotty Lee weight enumerators, Des. Codes Cryptogr. doi: 10.1007/s10623-012-9725-z.
[31] Sharma A., Sharma A. K., On MacWilliams type identities for r-fold joint m-spotty weight enumerators, Discrete Math. 312 (2012) 3316-3327.
[32] Siap I., The complete weight enumerator for codes over Mmxs(Fq), Lect. Notes Comput. Sci. (2001) 20-26.
[33] Siap I., A MacWilliams type identity, Turk. J. Math. 26 (2002) 465-473.
[34] Siap I., Ozen M., The complete weight enumerator for codes over Mnxs(R), Appl. Math. Lett. 17 (2004) 65-69.
[35] Siap I., MacWilliams identity for m-spotty Lee weight enumerator, Appl. Math. Lett. 23 (2010) 13-16.
[36] Siap I., An identity between the m-spotty weight enumerators of a linear code and its dual, Turk. J. Math. 36 (2012) 641-650.
[37] Siap V., Ozen M., MacWilliams identity for m-spotty Hamming weight enumerator over the ring F2 + ʋF2, Eur. J. Pure Appl. Math. 5 (2012) 373-379.
[38] Siap V., A MacWilliams type identity for m-spotty generalized Lee weight enumerators over Zq, Math. Sci. and Appl. E-Notes. 1 (2013) 111-116.
[39] Simonis J., MacWilliams identities and coordinate partitions, Linear Alg. Appl. 216 (1995) 81-91.
[40] Suzuki K., Kashiyama T., Fujiwara E., A general class of m-spotty byte error control codes, IEICE Trans. Fundam. E90-A (2007) 1418-1427.
[41] Suzuki K., Fujiwara E., MacWilliams identity for m-spotty weight enumerator, IEICE Trans. Fundam. E93-A (2010) 526-531.
[42] Suzuki K., Complete m-spotty weight enumerators of binary codes, Jacobi forms, and partial Epstein zeta functions, Discrete Math. 312 (2012) 265-278.
[43] Umanesan G., Fujiwara E., A class of random multiple bits in a byte error correcting and single byte error detecting (St/bEC-SbED) codes, IEEE Trans. Comput. 52 (2003) 835-847.
[44] Umanesan G.,Fujiwara E., A class of codes for correcting single spotty byte errors, IEICE Trans. Fundam. E86-A (2003) 704-714.
[45] Wood J., Duality for modules over _nite rings and applications to coding theory, Amer. J. Math., (1999) 555-575.
[46] Zaragoza R. M., Fossorier M., Lin S., Imai H., Multilevel coded modulation for unequal error protection and multistage decoding-Part I: Symmetric constellations, IEEE Trans. Commun. 48 (2000) 204-213.

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ISSN: 1304-7191
Yayın Aralığı: Yılda 4 Sayı
Yayıncı: Yıldız Teknik Üniversitesi

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