对RS和AG码新型软判决代数译码的研究

The algebraic decoding for Reed-Solomon (RS) and algebraic-geometric (AG) codes can correct errors beyond half of the code’s minimum Hamming distance. Hence, it has attracted many research interests. However, its high decoding complexity prevents a sooner implementation in industry. This is mainly caused by the interpolation process. Addressing this challenge, this project investigates a new interpolation approach, the module minimisation (MM). Based on MM, we will propose two low-complexity algebraic soft decoding algorithms, the algebraic Chase decoding (ACD) and the Koetter-Vardy (KV) decoding, namely the ACD-MM and the KV-MM algorithms. Re-encoding transform and progressive interpolation techniques will be further deployed to facilitate the ACD-MM and the KV-MM algorithms. The former reduces the MM interpolation complexity and the latter enables the decoding computation adapt to the quality of the received information. Moreover, this project also investigates powerful short-to-medium length codes to realise the modern communication vision of 'high transmission reliability and low energy consumption'. Utilising RS or AG codes, we can construct different structured codes that benefit a stronger decoding. This research may inspire an earlier industralisation of the algebraic decoding for RS and AG codes.

Reed-Solomon (RS) 和 Algebraic-Geometric (AG) 码的代数译码纠错能力可超过码最小汉明距离的一半，一直广受学术界的“青睐”。然而，其复杂的译码计算至今让工业界“望而却步”，这是由其插值运算导致的。针对此问题，本项目研究一种新型的插值方式 -- 模最小化 (Module Minimisation, MM)，并提出基于MM插值的代数 Chase 译码和 Koetter-Vardy (KV) 译码两种软判决算法。项目将进一步利用重编码和渐进插值等技术使这两种软判决代数译码“更轻便、更灵活”，前者能够进一步降低MM插值的复杂度，后者能够使译码计算根据接收信息的受干扰程度自适应调整。本项目还将研究性能优异的中短码，利用RS或AG码作为母码进行各种结构性编译码，满足现代通信对“高可靠、低能耗”的追求。本项目的研究可望为早日实现代数译码的工业化注入一剂“催化剂”。

本项目按计划完成了申请书所提出的Reed-Solomon (RS) 码和代数几何码的高效超限译码。本项目所取得的研究成果主要包括两个方面：第一、针对RS码的代数软判决译码 (algebraic soft decoding, ASD)，研究了一种以模基约简插值为基础的译码算法。为了降低插值复杂度，进一步引入了重编码变换 (re-encoding transform, ReT)算法。此外，引入了渐进的ASD和低复杂度Chase (low-complexity Chase, LCC)译码，使译码复杂度与信道条件相适应，从而促进了RS码的软译码过程。第二、针对代数几何码的代数译码进行了研究。其中包括引入ReT进行译码，设计椭圆码的ASD以及Hermitian码的LCC译码。上述研究成果已在该领域顶级期刊和会议上发表多篇论文，其中SCI论文10篇，国际会议论文12篇。本研究具有一定的实际意义，为RS码和代数几何码的高效译码提供了多种可靠的解决方案。它们在未来有被工业化应用的潜力。

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