A Family of Linear Codes and Their Subfield Codes
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摘要: 少重量线性码在秘密共享、强正则图、关联方案和认证码等方面有着广泛的应用。基于有限域上Kloosterman和，完全确定了一类$ q $元少重量线性码及其删余码的参数和重量分布，研究了它们的对偶码及其子域码，得到了关于球包界最优的线性码。
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关键词:
- 重量分布 /
- 子域码 /
- 删余码 /
- Kloosterman和
Abstract:Objective The study of the weight distributions of a linear code is very important in both theory and applications, since the weight distributions of a linear code can not only indicate the error-correcting ability of the code, but also allow the calculation of the error probability of error detection and correction. In addition, linear codes with few weights have many applications in secret sharing, strongly regular graphs, association schemes and authentication codes. Therefore, many scholars have focused their attention on the constructions of linear codes with few weights. Subfield codes of linear codes over finite fields have recently received much attention since they can produce optimal codes, which may have applications in data storage systems and communication systems. In recent years, subfield codes of linear codes over finite fields with good parameters have been widely studied. Motivates by the constructions, we choose a different defining set to extend their results. An objective of this paper is to study the weight distributions and the dual of this class of linear codes and their punctured codes. Another objective of this paper is to investigate their subfield codes to obtain some linear codes with few weights. Methods The most crucial step in researching a problem lies in the selection of the definition set. The calculation of the weight distributions of linear codes also relies on the decomposition of elements over finite fields into their subfields and the first four Pless power moments. Thanks to some known results on Kloosterman sums over finite fields, the lengths and weight distributions of this class of linear codes have a closed-form expression, and are completely determined in the binary case. The parameters of their duals are also determined which are optimal or almost optimal in the binary case. We present the trace representations of the subfield codes of this class of codes and their punctured codes, and then use the properties of characters on the finite field to determine their parameters and weight distributions and dualities of these subfield codes. Results and Discussions By selecting an appropriate defining set and based on Kloosterman sums over finite fields, the parameters and weight distributions of a family of q-ary linear codes with few weights and their punctured codes are completely determined. In addition, their dual codes and subfield codes are investigated, which are length-optimal and dimensional-optimal with respect to the Sphere-packing bound. A class of eight-weight linear codes and their puncture codes are also constructed as eight-weight linear codes. Their dual codes are all AMDS linear codes, and they are length-optimal and dimensional-optimal linear codes with respect to the Sphere-packing bound (see Theorems 1, 2 and Tables 1 ,2 ). The parameters and weight distributions of their subfield codes and their dual codes are provided (see Theorems 3 andTables 3 ). Besides, we also study the subfield codes of their punctured codes and determine the weight distributions and duality of the codes (see Theorems 4 andTables 4 ). Finally, all of these results have been verified by Magma through two examples.Conclusions This paper studies a family of q-ary linear codes with few weights and their punctured codes. Based on Kloosterman sums over finite fields, the weight distributions and parameters of the codes and their dual codes are determined, and the optimal linear codes with respect to the Sphere-packing bound are obtained. The weight distributions of their subfield codes and the parameters of their dual codes are determined, resulting in few-weight binary linear codes. -
Key words:
- weight distribution /
- subfield code /
- punctured code /
- Kloosterman sum
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表 1 $ {\mathcal{C}}_{\mathcal{D}} $的重量分布
重量 频数 $ 0 $ $ 1 $ $ {2}^{2m-1} $ $ {2}^{m}-1 $ $ {2}^{m-1}({2}^{m}-1)+1 $ $ {2}^{m}({2}^{m}-1) $ $ {2}^{m-1}({2}^{m}-1)-{1}/{2}(r-1)S-{1}/{2} $ $ {2}^{m-1}({2}^{m}-1) $ $ {2}^{m-1}({2}^{m}-1)+{1}/{2}(r-1)S+{1}/{2} $ $ ({2}^{m}-1)({2}^{m}-{2}^{m-1}) $ $ {2}^{m-1}({2}^{m}-1)-{1}/{2}(r-S-1) $ $ {2}^{m-2}({2}^{m}-1)(r-1)(r+1+S) $ $ {2}^{m-1}({2}^{m}-1)+{1}/{2}(1+r+S) $ $ {2}^{m-2}({2}^{m}-1)(r-1)(r+1-S) $ $ {2}^{m-1}({2}^{m}-1)-{1}/{2}(S-r-3) $ $ \begin{aligned} & ({2}^{m}-1)({2}^{m-1}-{2}^{m-2})\\& (r-1)(r+1+S)\end{aligned} $ $ {2}^{m-1}({2}^{m}-1)-{1}/{2}(S+r-3) $ $ \begin{aligned} & ({2}^{m}-1)({2}^{m-1}-{2}^{m-2})\\& (r-1)(r+1-S)\end{aligned} $ 
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表 2 $ \mathcal{C}_{\mathcal{D}}^{*} $的重量分布
重量 频数 $ 0 $ $ 1 $ $ {2}^{2m-1} $ $ {2}^{m}-1 $ $ {2}^{m-1}({2}^{m}-1) $ $ {2}^{m}({2}^{m}-1) $ $ {2}^{m-1}({2}^{m}-1)-{1}/{2}(r-1)S-{1}/{2} $ $ {2}^{m-1}({2}^{m}-1) $ $ {2}^{m-1}({2}^{m}-1)+{1}/{2}(r-1)S+{1}/{2} $ $ ({2}^{m}-1)({2}^{m}-{2}^{m-1}) $ $ {2}^{m-1}({2}^{m}-1)-{1}/{2}(1+r-S) $ $ {2}^{m-2}({2}^{m}-1)(r-1)(r+1+S) $ $ {2}^{m-1}({2}^{m}-1)-{1}/{2}(1-r-S) $ $ {2}^{m-2}({2}^{m}-1)(r-1)(r+1-S) $ $ {2}^{m-1}({2}^{m}-1)-{1}/{2}(S-r-1) $ $ ({2}^{m}-1)({2}^{m-1}-{2}^{m-2})(r-1)(r+1+S)$ $ {2}^{m-1}({2}^{m}-1)-{1}/{2}(S+r-1) $ $({2}^{m}-1)({2}^{m-1}-{2}^{m-2}) (r-1)(r+1-S)$
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表 3 $ \mathcal{C}_{\mathcal{D}}^{(2)} $的重量分布
重量 频数 $ 0 $ $ 1 $ $ {2}^{2m-1} $ $ 1 $ $ {2}^{2m-2} $ $ ({2}^{m}-2)({2}^{m}+1) $ $ {2}^{2m-2}+1 $ $ {2}^{m}({2}^{m}-1) $ $ {2}^{2m-2}-{2}^{m-2}-{2}^{m-2}(r-1)S $ $ 1 $ $ {2}^{2m-2}+{2}^{m-2}+{2}^{m-2}(r-1)S $ $ 1 $ $ {2}^{2m-2}-{2}^{m-2}(1+r-S) $ $ {(r-1)(r+1+S)}/{4}+{1}/{2} $ $ {2}^{2m-2}-{2}^{m-2}(1+r-S)+1 $ $ {(r-1)(r+1+S)}/{4}-{1}/{2} $ $ {2}^{2m-2}-{2}^{m-2}(1-r-S) $ $ {(r-1)(r+1-S)}/{4}-1 $ $ {2}^{2m-2}-{2}^{m-2}(1-r-S)+1 $ $ {(r-1)(r+1-S)}/{4}+1 $ $ {2}^{2m-2}+{2}^{m-2}(1+r-S) $ $ {(r-1)(r+1+S)}/{4}+{1}/{2} $ $ {2}^{2m-2}+{2}^{m-2}(1+r-S)+1 $ $ {(r-1)(r+1+S)}/{4}-{1}/{2} $ $ {2}^{2m-2}+{2}^{m-2}(1-r-S) $ $ {(r-1)(r+1-S)}/{4}-1 $ $ {2}^{2m-2}+{2}^{m-2}(1-r-S)+1 $ $ {(r-1)(r+1-S)}/{4}+1 $
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表 4 $ \mathcal{C}_{\mathcal{D}}^{*(2)} $的重量分布
重量 频数 $ 0 $ $ 1 $ $ {2}^{2m-1} $ $ 1 $ $ {2}^{2m-2} $ $ {2}^{m+1}({2}^{m}-1)-2 $ $ {2}^{2m-2}-{2}^{m-2}-{2}^{m-2}(r-1)S $ $ 1 $ $ {2}^{2m-2}+{2}^{m-2}+{2}^{m-2}(r-1)S $ $ 1 $ $ {2}^{2m-2}-{2}^{m-2}(1+r-S) $ $ {(r-1)(r+1+S)}/{2} $ $ {2}^{2m-2}-{2}^{m-2}(1-r-S) $ $ {(r-1)(r+1-S)}/{2} $ $ {2}^{2m-2}+{2}^{m-2}(1+r-S) $ $ frac{(r-1)(r+1+S)}/{2} $ $ {2}^{2m-2}+{2}^{m-2}(1-r-S) $ $ {(r-1)(r+1-S)}/{2} $
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