Weight Distributions of Some Classes of Irreducible Quasi-cyclic Codes of Index 2
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摘要: 少重量线性码在认证码、结合方案以及秘密共享方案的构造中有着重要的应用。如何构造少重量线性码一直是编码理论研究的重要内容。该文通过选取特殊的定义集，构造了有限域上指标为2的不可约拟循环码，利用有限域上的高斯周期确定了几类指标为2的不可约拟循环码的重量分布，并且得到了几类2-重量线性码和3-重量线性码。结果表明，由该文构造的3类2-重量线性码中有两类是极大距离可分(MDS)码，另一类达到了Griesmer界。Abstract: Few-weight linear codes have important applications in constructing authentication codes, association schemes and secret sharing schemes. How to construct few-weight linear codes has always been an important topic of coding theory. In this paper, irreducible quasi-cyclic codes of index 2 over finite fields are constructed by selecting a special defining set. The weight distribution of several classes of irreducible quasi-cyclic codes of index 2 are determined by using Gaussian periods over finite fields. Some classes of 2-weight linear codes and 3-weight linear codes are obtained. The results show that two of the three classes of 2-weight linear codes constructed in this paper are Maximum Distance Separable (MDS) codes and the other class reaches Griesmer bound.
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Key words:
- Linear codes /
- Quasi-cyclic codes /
- Gaussian periods /
- Weight distributions
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表 1 情形(1)：不可约拟循环码的重量分布
重量($ i $) 频数($ {A_i} $) 0 1 $ \dfrac{2}{{Nq}}(q - 1)({q^s} - 1 - 2\eta _0^{(2,{q^s})}) $ $ \dfrac{{{q^s} - 1}}{2} $ $ \dfrac{2}{{Nq}}(q - 1)({q^s} - 1 - 2\eta _1^{(2,{q^s})}) $ $ \dfrac{{{q^s} - 1}}{2} $ 
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表 2 情形(2)：不可约拟循环码的重量分布
重量($ i $) 频数($ {A_i} $) 0 1 $ \dfrac{2}{N}(q - 1){q^{s - 1}} $ $ {q^s} - 1 $
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表 3 情形(3)：不可约拟循环码的重量分布
重量($ i $) 频数($ {A_i} $) 0 1 $\dfrac{2}{ {Nq} }(q - 1)({q^s} - 1 - 3\eta _0^{(3,{q^s})})$ $ \dfrac{{{q^s} - 1}}{3} $ $ \dfrac{2}{{Nq}}(q - 1)({q^s} - 1 - 3\eta _1^{(3,{q^s})}) $ $ \dfrac{{{q^s} - 1}}{3} $ $ \dfrac{2}{{Nq}}(q - 1)({q^s} - 1 - 3\eta _2^{(3,{q^s})}) $ $ \dfrac{{{q^s} - 1}}{3} $
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表 4 情形(4)：不可约拟循环码的重量分布
重量($ i $) 频数($ {A_i} $) 0 1 $ \dfrac{1}{{Nq}}(q - 1)[2({q^s} - 1) - 3(\eta _0^{(3,{q^s})} + \eta _1^{(3,{q^s})})] $ $ \dfrac{{{q^s} - 1}}{3} $ $ \dfrac{1}{{Nq}}(q - 1)[2({q^s} - 1) - 3(\eta _1^{(3,{q^s})} + \eta _2^{(3,{q^s})})] $ $ \dfrac{{{q^s} - 1}}{3} $ $ \dfrac{1}{{Nq}}(q - 1)[2({q^s} - 1) - 3(\eta _2^{(3,{q^s})} + \eta _0^{(3,{q^s})})] $ $ \dfrac{{{q^s} - 1}}{3} $
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表 5 情形(5)：不可约拟循环码的重量分布
重量($ i $) 频数($ {A_i} $) 0 1 $ \dfrac{2}{{Nq}}(q - 1)({q^s} - 1 - 4\eta _0^{(4,{q^s})}) $ $ \dfrac{{{q^s} - 1}}{4} $ $ \dfrac{2}{{Nq}}(q - 1)({q^s} - 1 - 4\eta _1^{(4,{q^s})}) $ $ \dfrac{{{q^s} - 1}}{4} $ $ \dfrac{2}{{Nq}}(q - 1)({q^s} - 1 - 4\eta _2^{(4,{q^s})}) $ $ \dfrac{{{q^s} - 1}}{4} $ $ \dfrac{2}{{Nq}}(q - 1)({q^s} - 1 - 4\eta _3^{(4,{q^s})}) $ $ \dfrac{{{q^s} - 1}}{4} $
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表 6 情形(6)：不可约拟循环码的重量分布
重量($ i $) 频数($ {A_i} $) 0 1 $ \dfrac{2}{{Nq}}(q - 1)[({q^s} - 1) - 2(\eta _0^{(4,{q^s})} + \eta _1^{(4,{q^s})})] $ $ \dfrac{{{q^s} - 1}}{4} $ $ \dfrac{2}{{Nq}}(q - 1)[({q^s} - 1) - 2(\eta _1^{(4,{q^s})} + \eta _2^{(4,{q^s})})] $ $ \dfrac{{{q^s} - 1}}{4} $ $ \dfrac{2}{{Nq}}(q - 1)[({q^s} - 1) - 2(\eta _2^{(4,{q^s})} + \eta _3^{(4,{q^s})})] $ $ \dfrac{{{q^s} - 1}}{4} $ $ \dfrac{2}{{Nq}}(q - 1)[({q^s} - 1) - 2(\eta _3^{(4,{q^s})} + \eta _0^{(4,{q^s})})] $ $ \dfrac{{{q^s} - 1}}{4} $
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表 7 情形(7)：不可约拟循环码的重量分布
重量($ i $) 频数($ {A_i} $) 0 1 $ \dfrac{2}{{Nq}}(q - 1)[({q^s} - 1) - 2(\eta _0^{(4,{q^s})} + \eta _2^{(4,{q^s})})] $ $ \dfrac{{{q^s} - 1}}{2} $ $ \dfrac{2}{{Nq}}(q - 1)[({q^s} - 1) - 2(\eta _1^{(4,{q^s})} + \eta _3^{(4,{q^s})})] $ $ \dfrac{{{q^s} - 1}}{2} $
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[1] DING Cunsheng and YANG Jing. Hamming weights in irreducible cyclic codes[J]. Discrete Mathematics, 2013, 313(4): 434–446. doi: 10.1016/j.disc.2012.11.009 [2] BAE S, LI Chengju, and YUE Qin. On the complete weight enumerators of some reducible cyclic codes[J]. Discrete Mathematics, 2015, 338(12): 2275–2287. doi: 10.1016/j.disc.2015.05.016 [3] 管玥, 施敏加, 张欣, 等. 有限域上两类新的2-重量码的构造[J]. 电子学报, 2019, 47(3): 714–718. doi: 10.3969/j.issn.0372-2112.2019.03.028GUAN Yue, SHI Minjia, ZHANG Xin, et al. The construction of two new series of two-weight codes over finite fields[J]. Acta Electronica Sinica, 2019, 47(3): 714–718. doi: 10.3969/j.issn.0372-2112.2019.03.028 [4] SHI Minjia, ZHU Shixin, and YANG Shanlin. A class of optimal p-ary codes from one-weight codes over $ {F_p}[u]/\langle {u^m}\rangle $ [J]. Journal of The Franklin Institute, 2013, 350(5): 929–937. doi: 10.1016/j.jfranklin.2012.05.014[5] SHI Minjia, GUAN Yue, and SLOÉ P. Two new families of two-weight codes[J]. IEEE Transaction on Information Theory, 2017, 63(10): 6240–6246. doi: 10.1109/TIT.2017.2742499 [6] SHI Minjia, WU Rongsheng, LIU Yan, et al. Two and three weight codes over $ {F_p} + u{F_p} $ [J]. Cryptography and Communications, 2017, 9(5): 637–646. doi: 10.1007/s12095-016-0206-5[7] SHI Minjia and ZHANG Yiping. Quasi-twisted codes with constacyclic constituent codes[J]. Finite Fields and Their Applications, 2016, 39: 159–178. doi: 10.1016/j.ffa.2016.01.010 [8] SHI Minjia, QIAN Liqin, and SLOÉ P. On self-dual negacirculant codes of index two and four[J]. Designs, Codes and Cryptography, 2018, 86(11): 2485–2494. doi: 10.1007/s10623-017-0455-0 [9] 杜小妮, 吕红霞, 王蓉. 一类四重和六重线性码的构造[J]. 电子与信息学报, 2019, 41(12): 2995–2999. doi: 10.11999/JEIT180939DU Xiaoni, LÜ Hongxia, and WANG Rong. Construction of a class of linear codes with four-weight and six-weight[J]. Journal of Electronics &Information Technology, 2019, 41(12): 2995–2999. doi: 10.11999/JEIT180939 [10] DING Cunsheng. A class of three-weight and four-weight codes[M]. CHEE Y M, LI Chao, LING San, et al. Coding and Cryptology. IWCC 2009. Lecture Notes in Computer Science. Berlin: Springer, 2009: 34–42. [11] SCHMIDT B and WHITE C. All two-weight irreducible cyclic codes?[J]. Finite Fields and Their Applications, 2002, 8(1): 1–17. doi: 10.1006/ffta.2000.0293 [12] ZHOU Zhengchun and DING Cunsheng. Seven classes of three-weight cyclic codes[J]. IEEE Transactions on Communications, 2013, 61(10): 4120–4126. doi: 10.1109/TCOMM.2013.072213.130107 [13] BORGES J, FERNÁNDEZ-CÓRDOBA C, and TEN-VALLS R. $ {Z_2} $ -double cyclic codes[J]. Designs, Codes and Cryptography, 2018, 86(3): 463–479. doi: 10.1007/s10623-017-0334-8[14] GAO Jian, SHI Minjia, WU Tingting, et al. On double cyclic codes over $ {Z_4} $ [J]. Finite Fields and Their Applications, 2016, 39: 233–250. doi: 10.1016/j.ffa.2016.02.003[15] GAO Jian and HOU Xiaotong. $ {Z_4} $ -Double cyclic codes are asymptotically good[J]. IEEE Communications Letters, 2020, 24(8): 1593–1597. doi: 10.1109/LCOMM.2020.2992501[16] PATANKER N and SINGH S K. Weight distribution of a subclass of $ {Z_2} $ -double cyclic codes[J]. Finite Fields and Their Applications, 2019, 57: 287–308. doi: 10.1016/j.ffa.2019.03.003[17] MYERSON G. Period polynomials and Gauss sums for finite fields[J]. Acta Arithmetica, 1981, 39(3): 251–264. doi: 10.4064/aa-39-3-251-264 [18] DIAO Lingyu, GAO Jian, and LU Jiyong. Some results on $ {Z_p}{Z_p}[v] $ -additive cyclic codes[J]. Advances in Mathematics of Communications, 2020, 14(4): 557–572. doi: 10.3934/amc.2020029[19] HOU Xiaotong and GAO Jian. $ {Z_p}{Z_p}[v] $ -additive cyclic codes are asymptotically good[J]. Journal of Applied Mathematics and Computing, 2021, 66(1/2): 871–884. doi: 10.1007/s12190-020-01466-w -
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