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。该文基于有限域上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 weight distributions of linear codes is fundamental in both theory and applications. Weight distributions indicate the error-correcting capability of a code and allow the calculation of error probabilities for detection and correction. Linear codes with few weights also find applications in secret sharing, strongly regular graphs, association schemes, and authentication codes. Therefore, the construction of linear codes with few weights has attracted sustained attention. Subfield codes of linear codes over finite fields have recently received considerable interest because they can yield optimal codes with potential 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. Motivated by these constructions, a different defining set is selected to extend existing results. The objectives of this paper are to study the weight distributions and dual codes of this class of linear codes and their punctured codes, and to investigate their subfield codes to obtain linear codes with few weights. Methods The selection of the defining set is a key step in the analysis. The calculation of weight distributions relies on decomposing elements of finite fields into their subfields and applying the first four Pless power moments. Using known results on Kloosterman sums over finite fields, the lengths and weight distributions of this class of linear codes admit closed-form expressions and are completely determined in the binary case. The parameters of their dual codes are also determined and are optimal or almost optimal in the binary case. Trace representations of the subfield codes of this class of codes and their punctured codes are derived. Properties of characters over finite fields are then used to determine the parameters, weight distributions, and dualities of these subfield codes. Results and Discussions By selecting an appropriate defining set and using 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. Their dual codes and subfield codes are also examined and are shown to be length-optimal and dimension-optimal with respect to the Sphere-packing bound. A class of eight-weight linear codes and their punctured codes is constructed. The corresponding dual codes are all AMDS linear codes, and they are length-optimal and dimension-optimal linear codes with respect to the Sphere-packing bound (see Theorems 1 and 2, and Tables 1 and2 ). The parameters and weight distributions of their subfield codes and the corresponding dual codes are provided (see Theorem 3 and Table 3). In addition, the subfield codes of the punctured codes are studied, and the weight distributions and duality of these codes are determined (see Theorem 4 andTable 4 ). All results are verified using Magma through two examples.Conclusions A family of q-ary linear codes with few weights and their punctured codes is studied. Based on Kloosterman sums over finite fields, the weight distributions and parameters of the codes and their dual codes are determined, yielding optimal linear codes with respect to the Sphere-packing bound. The weight distributions of their subfield codes and the parameters of the corresponding dual codes are also 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) $ $ (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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