{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,22]],"date-time":"2024-07-22T10:36:11Z","timestamp":1721644571264},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2013,10,4]],"date-time":"2013-10-04T00:00:00Z","timestamp":1380844800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2015,3]]},"DOI":"10.1007\/s10623-013-9883-7","type":"journal-article","created":{"date-parts":[[2013,10,3]],"date-time":"2013-10-03T02:50:29Z","timestamp":1380768629000},"page":"681-701","source":"Crossref","is-referenced-by-count":6,"title":["On generator matrices and parity check matrices of generalized integer codes"],"prefix":"10.1007","volume":"74","author":[{"given":"Hajime","family":"Matsui","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2013,10,4]]},"reference":[{"key":"9883_CR1","doi-asserted-by":"crossref","unstructured":"Bhaintwal M., Wasan S.K.: On quasi-cyclic codes over $$Z_q$$ Z q . Appl. Algebra Eng. Commun. Comput. 20(5\u20136), 459\u2013480 (2009).","DOI":"10.1007\/s00200-009-0110-8"},{"key":"9883_CR2","doi-asserted-by":"crossref","unstructured":"Blake I.: Codes over certain rings. Inf. Control 20(4), 396\u2013404 (1972).","DOI":"10.1016\/S0019-9958(72)90223-9"},{"key":"9883_CR3","doi-asserted-by":"crossref","unstructured":"Blake I.: Codes over integer residue rings. Inf. Control 29(4), 295\u2013300 (1975).","DOI":"10.1016\/S0019-9958(75)80001-5"},{"key":"9883_CR4","doi-asserted-by":"crossref","unstructured":"Calderbank A.R., Sloane N.J.A.: Modular and $$p$$ p -adic cyclic codes. Des. Codes Cryptogr. 6(1), 21\u201335 (1995).","DOI":"10.1007\/BF01390768"},{"key":"9883_CR5","doi-asserted-by":"crossref","unstructured":"Cao Y.: Structural properties and enumeration of 1-generator generalized quasi-cyclic codes. Des. Codes Cryptogr. 60(1), 67\u201369 (2010).","DOI":"10.1007\/s10623-010-9417-5"},{"key":"9883_CR6","unstructured":"Chen L., Xu J., Djurdjevic I., Lin S.: Near-Shannon-limit quasi-cyclic low-density parity-check codes. IEEE Trans. Commun. 52(7), 1038\u20131042 (2004)."},{"key":"9883_CR7","unstructured":"Esmaeili M., Yari S.: Generalized quasi-cyclic codes: structural properties and code construction. Appl. Algebra Eng. Commun. Comput. 20(2), 159\u2013173 (2009)."},{"key":"9883_CR8","unstructured":"Fossorier M.P.C.: Quasi-cyclic low density parity check codes from circulant permutation matrices. IEEE Trans. Inf. Theory 50(8), 1788\u20131793 (2004)."},{"key":"9883_CR9","unstructured":"Han Vinck A.J., Morita H.: Codes over the ring of integers modulo m. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E81-A(10), 2013\u20132018 (1998)."},{"key":"9883_CR10","unstructured":"Heegard C., Little J., Saints K.: Systematic encoding via Gr\u00f6bner bases for a class of algebraic geometric Goppa codes. IEEE Trans. Inf. Theory 41(6), 1752\u20131761 (1995)."},{"key":"9883_CR11","doi-asserted-by":"crossref","unstructured":"Huffman W.C.: On the classification and enumeration of self-dual codes. Finite Fields Appl. 11(3), 451\u2013490 (2005).","DOI":"10.1016\/j.ffa.2005.05.012"},{"key":"9883_CR12","unstructured":"Kamiya N.: High-rate quasi-cyclic low-density parity-check codes derived from finite affine planes. IEEE Trans. Inf. Theory 53(4), 1444\u20131459 (2007)."},{"key":"9883_CR13","doi-asserted-by":"crossref","unstructured":"Kamiya N., Fossorier M.P.C.: Quasi-cyclic codes from a finite affine plane. Des. Codes Cryptogr. 38(3), 311\u2013329 (2006).","DOI":"10.1007\/s10623-005-6685-6"},{"key":"9883_CR14","unstructured":"Kostadinov H., Morita H., Manev N.: Integer codes correcting single errors of specific types $$(\\pm e_1,\\pm e_2,...,\\pm e_s)$$ ( \u00b1 e 1 , \u00b1 e 2 , . . . , \u00b1 e s ) . IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E86-A(7), 1843\u20131849 (2003)."},{"key":"9883_CR15","unstructured":"Kostadinov H., Morita H., Manev N.: Derivation on bit error probability of coded QAM using integer codes. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E87-A(12), 3397\u20133403 (2004)."},{"key":"9883_CR16","doi-asserted-by":"crossref","unstructured":"Kostadinov H., Morita H., Iijima N., Han Vinck A.J., Manev N.: Soft decoding of integer codes and their application to coded modulation. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E93-A(7), 1363\u20131370 (2010).","DOI":"10.1587\/transfun.E93.A.1363"},{"key":"9883_CR17","unstructured":"Kou Y., Lin S., Fossorier M.P.C.: Low density parity check codes based on finite geometries: a rediscovery and new results. IEEE Trans. Inf. Theory 47(7), 2711\u20132761 (2001)."},{"key":"9883_CR18","doi-asserted-by":"crossref","unstructured":"Lally K., Fitzpatrick P.: Algebraic structure of quasi-cyclic codes. Discret. Appl. Math. 111(1\u20132), 157\u2013175 (2001).","DOI":"10.1016\/S0166-218X(00)00350-4"},{"key":"9883_CR19","unstructured":"Lin S., Costello D.J.: Error Control Coding: Fundamentals and Applications, 2nd edn. Prentice-Hall, Englewood Cliffs (2004)."},{"key":"9883_CR20","unstructured":"Ling S., Sol\u00e9 P.: On the algebraic structure of quasi-cyclic codes I: finite fields. IEEE Trans. Inf. Theory 47(7), 2751\u20132760 (2001)."},{"key":"9883_CR21","doi-asserted-by":"crossref","unstructured":"Ling S., Sol\u00e9 P.: On the algebraic structure of quasi-cyclic codes II: chain rings. Des. Codes Cryptogr. 30(1), 113\u2013130 (2003).","DOI":"10.1023\/A:1024715527805"},{"key":"9883_CR22","unstructured":"Ling S., Sol\u00e9 P.: On the algebraic structure of quasi-cyclic codes III: generator theory. IEEE Trans. Inf. Theory 51(7), 2692\u20132700 (2005)."},{"key":"9883_CR23","doi-asserted-by":"crossref","unstructured":"Ling S., Niederreiter H., Sol\u00e9 P.: On the algebraic structure of quasi-cyclic codes IV: repeated roots. Des. Codes Cryptogr. 38(3), 337\u2013361 (2006).","DOI":"10.1007\/s10623-005-1431-7"},{"key":"9883_CR24","unstructured":"MacWilliams F.J., Sloane N.J.A.: The Theory of Error Correcting Codes, 9th edn. Elsevier, North Holland (1988)."},{"key":"9883_CR25","unstructured":"Matsui H.: On polynomial generator matrices of generalized quasi-cyclic codes. In: 6th Asia-Europe Workshop on Information Theory, Ishigaki Island, Okinawa, Japan, 22\u201324 Oct 2010."},{"key":"9883_CR26","unstructured":"Nakamura K.: A class of error-correcting codes for DPSK channels. In: IEEE International Conference on Communications, Boston, MA, June 10\u201314 1979, pp. 45.4.1\u201345.4.5."},{"key":"9883_CR27","unstructured":"Nebe G., Rains E.M., Sloane N.J.A.: Self-Dual Codes and Invariant Theory. Springer, Berlin (2006)."},{"key":"9883_CR28","unstructured":"Roth R.M.: Introduction to Coding Theory. Cambridge University Press, Cambridge (2005)."},{"key":"9883_CR29","unstructured":"S\u00e9guin G.E.: A class of 1-generator quasi-cyclic codes. IEEE Trans. Inf. Theory 50(8), 1745\u20131753 (2004)."},{"key":"9883_CR30","doi-asserted-by":"crossref","unstructured":"Serre J.-P.: A Course in Arithmetic. Springer, Berlin (1973).","DOI":"10.1007\/978-1-4684-9884-4"},{"key":"9883_CR31","unstructured":"Shimura G.: Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton (1971)."},{"key":"9883_CR32","unstructured":"Siap I., Kulhan N.: The structure of generalized quasi-cyclic codes. Appl. Math. E-Notes 5, 24\u201330 (2005)."},{"key":"9883_CR33","unstructured":"Sol\u00e9 P. (ed.): Codes over Rings. In: Proceedings of the Cimpa Summer School, Ankara, Turkey, 18\u201329 Aug 2008."},{"key":"9883_CR34","doi-asserted-by":"crossref","unstructured":"Spiegel E.: Codes over $$\\text{ Z }_m$$ Z m . Inf. Control 35(1), 48\u201351 (1977).","DOI":"10.1016\/S0019-9958(77)90526-5"},{"key":"9883_CR35","unstructured":"Tang H., Xu J., Lin S., Abdel-Ghaffar K.A.S.: Codes on finite geometries. IEEE Trans. Inf. Theory 51(2), 572\u2013596 (2005)."},{"key":"9883_CR36","doi-asserted-by":"crossref","unstructured":"Van V.T., Matsui H., Mita S.: Computation of Gr\u00f6bner basis for systematic encoding of generalized quasi-cyclic codes. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E92-A(9), 2345\u20132359 (2009).","DOI":"10.1587\/transfun.E92.A.2345"},{"key":"9883_CR37","unstructured":"Van V.T., Matsui H., Mita S.: A class of generalized quasi-cyclic LDPC codes: high-rate and low-complexity encoder for data storage devices. In: IEEE Global Communications Conference (GLOBECOM), Miami, FL, 2010, pp. 6\u201310."}],"container-title":["Designs, Codes and Cryptography"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-013-9883-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10623-013-9883-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-013-9883-7","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T15:58:37Z","timestamp":1559231917000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10623-013-9883-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,10,4]]},"references-count":37,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2015,3]]}},"alternative-id":["9883"],"URL":"https:\/\/doi.org\/10.1007\/s10623-013-9883-7","relation":{},"ISSN":["0925-1022","1573-7586"],"issn-type":[{"value":"0925-1022","type":"print"},{"value":"1573-7586","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,10,4]]}}}