Gao, You; Wang, Gang Bounds on subspace codes based on subspaces of type \((m, 1)\) in singular linear space. (English) Zbl 1463.94069 J. Appl. Math. 2014, Article ID 497958, 9 p. (2014). Summary: The sphere-packing bound, Singleton bound, Wang-Xing-Safavi-Naini bound, Johnson bound, and Gilbert-Varshamov bound on the subspace codes \((n+l,M,d,(m,1))_q\) based on subspaces of type \((m,1)\) in singular linear space \(\mathbb{F}_q^{(n+l)}\) over finite fields \(\mathbb{F}_q\) are presented. Then, we prove that codes based on subspaces of type \((m,1)\) in singular linear space attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in \(\mathbb{F}_q^{(n+l)}\). Cited in 3 Documents MSC: 94B65 Bounds on codes 94A62 Authentication, digital signatures and secret sharing 94B05 Linear codes (general theory) 94A05 Communication theory 94B60 Other types of codes PDF BibTeX XML Cite \textit{Y. Gao} and \textit{G. Wang}, J. Appl. Math. 2014, Article ID 497958, 9 p. (2014; Zbl 1463.94069) Full Text: DOI References: [1] Koetter, R.; Kschischang, F. R., Coding for errors and erasures in random network coding, IEEE Transactions on Information Theory, 54, 8, 3579-3591 (2008) · Zbl 1318.94111 [2] Ahlswede, R.; Cai, N.; Li, S. R. a., Network information flow, IEEE Transactions on Information Theory, 46, 4, 1204-1216 (2000) · Zbl 0991.90015 [3] Gadouleau, M.; Yan, Z., Packing and covering properties of subspace codes for error control in random linear network coding, IEEE Transactions on Information Theory, 56, 5, 2097-2108 (2010) · Zbl 1366.94679 [4] Koetter, R.; Kschischang, F. R., Error correction in random network, Proceedings of the 2nd Annual Workshop on Information Theory and Applications [5] Ho, T.; Médard, M.; Koetter, R.; Karger, D.; Effros, M.; Shi, J.; Leong, B., A random linear network coding approach to multicast, IEEE Transactions on Information Theory, 52, 10, 4413-4430 (2006) · Zbl 1323.94010 [6] Silva, D.; Kschischang, F. R.; Koetter, R., A rank-metric approach to error control in random network coding, IEEE Transactions on Information Theory, 54, 9, 3951-3967 (2008) · Zbl 1318.94119 [7] Gabidulin, E. M.; Bossert, M., Codes for network coding, Proceedings of the IEEE International Symposium on Information Theory (ISIT ’08) [8] Gadouleau, M.; Yan, Z., Constant-rank codes, Proceedings of the IEEE International Symposium on Information Theory (ISIT ’08) [9] MacWilliams, F. J.; Sloane, N. J. A., The Theory of Error-Correcting Codes (1977), Amsterdam, The Netherlands: North-Holland, Amsterdam, The Netherlands · Zbl 0369.94008 [10] Wang, H.; Xing, C.; Safavi-Naini, R., Linear authentication codes: bounds and constructions, IEEE Transactions on Information Theory, 49, 4, 866-872 (2003) · Zbl 1063.94087 [11] Etzion, T.; Vardy, A., Error-correcting codes in projective space, IEEE Transactions on Information Theory, 57, 2, 1165-1173 (2011) · Zbl 1366.94589 [12] Xia, S. T.; Fu, F. W., Johnson type bounds on constant dimension codes, Designs, Codes and Cryptography, 50, 2, 163-172 (2009) · Zbl 1237.94151 [13] Schwartz, M.; Etzion, T., Codes and anticodes in the Grassman graph, Journal of Combinatorial Theory A, 97, 1, 27-42 (2002) · Zbl 1006.94028 [14] Etzion, T., Perfect byte-correcting codes, IEEE Transactions on Information Theory, 44, 7, 3140-3146 (1998) · Zbl 0953.94027 [15] Wang, K.; Guo, J.; Li, F., Singular linear space and its applications, Finite Fields and Their Applications, 17, 5, 395-406 (2011) · Zbl 1234.51008 [16] Wan, Z., Geometry of Classical Groups Over Finite Fields (2002), Beijing, China: Science Press, Beijing, China This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.