[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86064-en":3,"doc-seo-86064-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86064,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Faster Closest-Point Algorithms for the E6 and E7 Lattices","Dual lattices E6* and E7* are key for source coding and data compression because they achieve the smallest normalized second moments in six and seven dimensions, minimizing average quantization error. Efficient nearest-lattice-point decoding is required for practical use. The classical coset-decoding method uses multiple coset decodings and explicit distance computations. This paper collapses all coset decodings into a single sweep via glue-vector reformulation, reducing operation counts substantially and enabling further constant-factor improvements, with an open sort-free linear-time question.","arXiv :2607 . 10885v 1 [ cs .IT] 12 Jul 2026  \nFaster Closest-Point Algorithms  \nfor the E and E Lattices  \nYuriy Reznik  \nMassachusetts Institute of Technology  \n[yreznik@mit.edu](yreznik@mit.edu)  \nAbstract  \nThe 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 E6 and E∗6 by Takizawa, Yagi, and Kawabata (TYK), decodes these lattices as unions of cosets of root lattices An: 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.  \nThis 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 ⊕ 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 An 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 × reduction for E∗6 and 3–4× 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.  \n1 Introduction  \n1.1 Motivation  \nA central figure of merit of a lattice used for quantization is its normalized second moment G, which measures the average squared quantization error per dimension incurred when a uniformly distributed source is quantized to the nearest lattice point. In dimensions six and seven, the best lattice quantizers known are the dual lattices E∗6 and E∗7 [3, Ch. 2, Table 2.3]: their normalized second moments, computed exactly by Worley [4, 5], are the smallest among all known six- and seven-dimensional lattices. Consequently, whenever an application calls for vector quantization in these dimensions—examples include gain/shape and subband quantizers in audio coding, quantization of short feature or parameter blocks in learned models, and coded modulation over six-or seven-dimensional constellations—E∗6 and E∗7 are the natural first candidates.  \nA theoretically good lattice is useful in practice only if the closest-point problem—given a query y, find x∗ = arg minx∈Λ ∥y − x∥2—can be solved quickly. This paper presents faster closest-point algorithms for E∗6 and E∗7; the same technique also covers E6 and E7 .  \n1.2 Prior work  \nConway and Sloane [1] gave the classical fast closest-point algorithms for the root lattices Zn , An , Dn , E7 , E8 and their duals. Their central device is coset decoding: if a lattice of interest can be written as a finite union  \nd−1 Λ′ = [(ri+ Λ)  \ni=0  \nof shifted copies (cosets) of a lattice Λ for which a closest-point algorithm is available, then Λ′ is decoded by quantizing y − ri to Λ for each i, shifting each result back by ri, and keeping the closest of the d candidates. For example, E∗7 is the union of d = 4 cosets of A7 and is decoded by four A7 quantizations plus four distance computations. Soft-decision variants of these decoders appear in [2] . Notably, [1] does not treat E6 and E∗6 .  \nThat gap was closed by Takizawa, Yagi, and Kawabata (TYK) [9], who also generalized the optimality proofs of the Conway–Sloane quantizers to ℓp norms. TYK showed that, in the standard coordinates E6 ⊂ R8 , the lattice splits across two orthogonal coordinate blocks into a","cbCainAhc5Zli330","https://ap.wps.com/l/cbCainAhc5Zli330","pdf",276923,2,1,9,"English","en",105,"# Abstract\n# Introduction\n## Motivation\n## Prior work\n# Preliminaries\n## The lattices An, A*n, and glue vectors","[{\"question\":\"Why are E6* and E7* important in coding and compression?\",\"answer\":\"They are dual lattices whose normalized second moments are the smallest among known six- and seven-dimensional lattices, yielding the lowest average quantization error for nearest-point quantization.\"},{\"question\":\"What is the limitation of the classical Conway–Sloane/TYK coset-decoding approach for E6* and E7*?\",\"answer\":\"It decodes each coset separately and requires multiple coset decodings plus explicit distance computations, leading to higher computational cost.\"},{\"question\":\"How does this paper speed up closest-point decoding for E6* and E7*?\",\"answer\":\"It reformulates the TYK decompositions using glue vectors and shows that a candidate chain sweep (from MCQ for A*n) visits each glue class exactly once, allowing all glue cosets to be decoded simultaneously with roughly the cost of a single A*n quantization.\"}]",1784208234,23,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"faster-closest-point-algorithms-for-the-e6-and-e7-lattices","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/faster-closest-point-algorithms-for-the-e6-and-e7-lattices/86064/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why are E6* and E7* important in coding and compression?","Question",{"text":75,"@type":76},"They are dual lattices whose normalized second moments are the smallest among known six- and seven-dimensional lattices, yielding the lowest average quantization error for nearest-point quantization.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the limitation of the classical Conway–Sloane/TYK coset-decoding approach for E6* and E7*?",{"text":80,"@type":76},"It decodes each coset separately and requires multiple coset decodings plus explicit distance computations, leading to higher computational cost.",{"name":82,"@type":73,"acceptedAnswer":83},"How does this paper speed up closest-point decoding for E6* and E7*?",{"text":84,"@type":76},"It reformulates the TYK decompositions using glue vectors and shows that a candidate chain sweep (from MCQ for A*n) visits each glue class exactly once, allowing all glue cosets to be decoded simultaneously with roughly the cost of a single A*n 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