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May  2016, 10(2): 367-377. doi: 10.3934/amc.2016011

## The geometric structure of relative one-weight codes

 1 Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, China 2 Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan, Hubei 430062, China

Received  July 2014 Revised  August 2015 Published  April 2016

The geometric structure of any relative one-weight code is determined, and by using this geometric structure, the support weight distribution of subcodes of any relative one-weight code is presented. An application of relative one-weight codes to the wire-tap channel of type II with multiple users is given, and certain kinds of relative one-weight codes all of whose nonzero codewords are minimal are determined.
Citation: Zihui Liu, Xiangyong Zeng. The geometric structure of relative one-weight codes. Advances in Mathematics of Communications, 2016, 10 (2) : 367-377. doi: 10.3934/amc.2016011
##### References:
 [1] A. Ashikhmin and A. Barg, Minimal vectors in linear codes,, IEEE Trans. Inf. Theory, 44 (1998), 2010. doi: 10.1109/18.705584. Google Scholar [2] W. D. Chen and T. Kløve, The weight hierarchies of q-ary codes of dimension 4,, IEEE Trans. Inf. Theory, 42 (1996), 2265. doi: 10.1109/18.556621. Google Scholar [3] Z. H. Liu and W. D. Chen, Notes on the value function,, Des. Codes Crypt., 54 (2010), 11. doi: 10.1007/s10623-009-9305-z. Google Scholar [4] Z. H. Liu, W. D. Chen, Z. M. Sun and X. Y. Zeng, Further results on support weights of certain subcodes,, Des. Codes Crypt., 61 (2011), 119. doi: 10.1007/s10623-010-9442-4. Google Scholar [5] Z. H. Liu and X. W. Wu, On relative constant-weight codes,, Des. Codes Crypt., 75 (2015), 127. doi: 10.1007/s10623-013-9896-2. Google Scholar [6] Y. Luo, C. Mitrpant, A. J. H. Vinck and K. Chen, Some new characters on the wire-tap channel of type II,, IEEE Trans. Inf. Theory, 51 (2005), 1222. doi: 10.1109/TIT.2004.842763. Google Scholar [7] F. J. MacWilliams, N. J. A. Sloane, The Theory of Error Correcting Codes,, North Holland, (1977). Google Scholar [8] V. K. Wei, Generalized Hamming weight for linear codes,, IEEE Trans. Inf. Theory, 37 (1991), 1412. doi: 10.1109/18.133259. Google Scholar [9] J. A. Wood, Relative one-weight linear codes,, Des. Codes Crypt., 72 (2014), 331. doi: 10.1007/s10623-012-9769-0. Google Scholar [10] J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes,, IEEE Trans. Inf. Theory, 52 (2006), 206. doi: 10.1109/TIT.2005.860412. Google Scholar

show all references

##### References:
 [1] A. Ashikhmin and A. Barg, Minimal vectors in linear codes,, IEEE Trans. Inf. Theory, 44 (1998), 2010. doi: 10.1109/18.705584. Google Scholar [2] W. D. Chen and T. Kløve, The weight hierarchies of q-ary codes of dimension 4,, IEEE Trans. Inf. Theory, 42 (1996), 2265. doi: 10.1109/18.556621. Google Scholar [3] Z. H. Liu and W. D. Chen, Notes on the value function,, Des. Codes Crypt., 54 (2010), 11. doi: 10.1007/s10623-009-9305-z. Google Scholar [4] Z. H. Liu, W. D. Chen, Z. M. Sun and X. Y. Zeng, Further results on support weights of certain subcodes,, Des. Codes Crypt., 61 (2011), 119. doi: 10.1007/s10623-010-9442-4. Google Scholar [5] Z. H. Liu and X. W. Wu, On relative constant-weight codes,, Des. Codes Crypt., 75 (2015), 127. doi: 10.1007/s10623-013-9896-2. Google Scholar [6] Y. Luo, C. Mitrpant, A. J. H. Vinck and K. Chen, Some new characters on the wire-tap channel of type II,, IEEE Trans. Inf. Theory, 51 (2005), 1222. doi: 10.1109/TIT.2004.842763. Google Scholar [7] F. J. MacWilliams, N. J. A. Sloane, The Theory of Error Correcting Codes,, North Holland, (1977). Google Scholar [8] V. K. Wei, Generalized Hamming weight for linear codes,, IEEE Trans. Inf. Theory, 37 (1991), 1412. doi: 10.1109/18.133259. Google Scholar [9] J. A. Wood, Relative one-weight linear codes,, Des. Codes Crypt., 72 (2014), 331. doi: 10.1007/s10623-012-9769-0. Google Scholar [10] J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes,, IEEE Trans. Inf. Theory, 52 (2006), 206. doi: 10.1109/TIT.2005.860412. Google Scholar
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