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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