AI 中文总结
研究聚焦于源编码和数据压缩应用中对偶格$E_6^*$和$E_7^*$，已知方法解码需多次陪集解码与距离计算，本文改进算法，将陪集解码合并为一次扫描，大幅降低计算量。
AI 中文摘要
对偶格$E_6^*$和$E_7^*$在源编码和数据压缩应用中特别受关注。在所有已知的六维和七维格中，它们具有最小的归一化二阶矩，即最小的平均量化误差。在实践中使用它们需要快速的最近点（最近格点）算法。已知的方法，由Conway和Sloane提出，并由Takizawa、Yagi和Kawabata（TYK）对$E_6$和$E_6^*$进行了完善，将这些格解码为根格$A_n$的陪集的并集：每个陪集单独解码，并保留最佳结果。对于$E_7^*$需要四次陪集解码，对于$E_6^*$需要六次陪集解码，同时还需要显式的距离计算。本文表明所有这些陪集解码可以合并为一次扫描。用胶合向量重新表述，TYK分解表明$E_7^*$是$A_7^*$的偶胶合类的并集，并且$E_6^*$是$A_1^*\oplus A_5^*$的奇偶匹配子格。由McKilliam、Clarkson和Quinn（MCQ）为$A_n^*$的最近点算法构造候选链恰好访问$A_n$的每个胶合类一次，并且在每个类中是最优的。因此，每个坐标块进行一次排序扫描就可以同时得到所有胶合陪集的最近点，并且以大约一次$A_5^*$或$A_7^*$量化的成本对$E_6^*$和$E_7^*$进行解码。粗略的运算次数表明，相对于逐个陪集解码，$E_6^*$的计算量减少了4 - 6倍，$E_7^*$减少了3 - 4倍。我们还讨论了从$A_n^*$算法的最新改进中可获得的进一步的常数因子改进，以及一个关于无排序线性时间解码的开放问题。
英文摘要
The dual lattices $E_6^*$ and $E_7^*$ are of particular interest in source coding and data compression applications. Among all known lattices in dimensions six and seven they attain the smallest normalized second moments, i.e., the smallest average quantization error. Their use in practice requires fast closest-point (nearest-lattice-point) algorithms. The known approach, due to Conway and Sloane and completed for $E_6$ and $E_6^*$ by Takizawa, Yagi, and Kawabata (TYK), decodes these lattices as unions of cosets of root lattices $A_n$: each coset is decoded separately, and the best result is kept. This requires four coset decodings for $E_7^*$ and six for $E_6^*$, together with explicit distance computations. This paper shows that all these coset decodings can be collapsed into a single sweep. Reformulated in terms of glue vectors, the TYK decompositions state that $E_7^*$ is the union of the even glue classes of $A_7^*$, and that $E_6^*$ is a parity-matched sublattice of $A_1^*\oplus A_5^*$. The candidate chain constructed by the closest-point algorithm of McKilliam, Clarkson, and Quinn (MCQ) for $A_n^*$ visits every glue class of $A_n$ exactly once and is optimal within each class. Consequently, one sorted sweep per coordinate block yields the closest points of all glue cosets simultaneously, and $E_6^*$ and $E_7^*$ are decoded at roughly the cost of a single $A_5^*$ or $A_7^*$ quantization. Rough operation counts indicate a $4$--$6\times$ reduction for $E_6^*$ and $3$--$4\times$ for $E_7^*$ relative to coset-by-coset decoding. We also discuss further constant-factor improvements available from recent refinements of the $A_n^*$ algorithms, and an open question concerning sort-free linear-time decoding.
