{"id":1190061,"date":"2026-10-16T02:16:08","date_gmt":"2026-10-16T09:16:08","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=1190061"},"modified":"2026-10-16T02:22:11","modified_gmt":"2026-10-16T09:22:11","slug":"nonlinear-fixed-distance-codes-via-real-lifts-of-moment-configurations","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/nonlinear-fixed-distance-codes-via-real-lifts-of-moment-configurations\/","title":{"rendered":"Nonlinear fixed-distance codes via real lifts of moment configurations"},"content":{"rendered":"\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let <math><mrow><msub><mi>A<\/mi><mi>q<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>,<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math> denote the largest size of a code of length <math><mi>n<\/mi><\/math> and minimum Hamming distance <math><mi>d<\/mi><\/math> over an alphabet of <math><mi>q<\/mi><\/math> symbols. For fixed <math><mrow><mi>q<\/mi><mo>\u2265<\/mo><mn>2<\/mn><\/mrow><\/math> and <math><mrow><mi>d<\/mi><mo>\u2265<\/mo><mn>4<\/mn><\/mrow><\/math> we construct codes with redundancy<br><math><mrow><mi>n<\/mi><mo>&#8211;<\/mo><msub><mo>log<\/mo><mi>q<\/mi><\/msub><msub><mi>A<\/mi><mi>q<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>,<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mo stretchy=\"false\">(<\/mo><mi>d<\/mi><mo>&#8211;<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><msub><mo>log<\/mo><mi>q<\/mi><\/msub><mi>n<\/mi><mo>+<\/mo><mi>O<\/mi><mo stretchy=\"false\">(<\/mo><msqrt><mrow><mrow><mo>log<\/mo><mi>n<\/mi><\/mrow><\/mrow><\/msqrt><mo stretchy=\"false\">)<\/mo><\/mrow><\/math> for all large <math><mi>n<\/mi><\/math>.<br>No algebraic structure is imposed on the alphabet, and the codes are not linear. For <math><mrow><mi>d<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><\/math> and <math><mrow><mi>d<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><\/math> the coefficient of <math><mrow><msub><mo>log<\/mo><mi>q<\/mi><\/msub><mi>n<\/mi><\/mrow><\/math> matches the Hamming bound, so the redundancy is determined up to lower-order terms. At distance four the codes are provably better than every linear code: by the polynomial-method bound for cap sets, <math><mrow><msub><mi>\ud835\udd3d<\/mi><mi>q<\/mi><\/msub><\/mrow><\/math>-linear codes with <math><mrow><mi>q<\/mi><mo>\u2265<\/mo><mn>3<\/mn><\/mrow><\/math> require a strictly larger coefficient in front of <math><mrow><msub><mo>log<\/mo><mi>q<\/mi><\/msub><mi>n<\/mi><\/mrow><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The construction imposes moment checksums over an auxiliary finite field, as BCH codes do, but one fewer than the distance would normally require. The affine-line theorem from prior work describes the dependencies left by these checks. We exclude them by placing the code positions inside a specially structured subset of the field, built from Behrend&#8217;s sphere construction with the help of algebraic number theory.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let Aq(n,d) denote the largest size of a code of length n and minimum Hamming distance d over an alphabet of q symbols. For fixed q\u22652 and d\u22654 we construct codes with redundancyn&#8211;logqAq(n,d)\u2264(d&#8211;3)logqn+O(logn) for all large n.No algebraic structure is imposed on the alphabet, and the codes are not linear. For d=4 and d=5 the [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"user_nicename","value":"Sergey 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