MinT - Best Known (83−72, 83, s)-Nets in Base 25

Best Known (83−72, 83, s)-Nets in Base 25

(83−72, 83, 126)-Net over F₂₅ — Constructive and digital

Digital (11, 83, 126)-net over F₂₅, using

t-expansion [i] based on digital (10, 83, 126)-net over F₂₅, using
- net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
    - the Hermitian function field over F₂₅ [i]

(83−72, 83, 974)-Net over F₂₅ — Upper bound on s (digital)

There is no digital (11, 83, 975)-net over F₂₅, because

extracting embedded orthogonal array [i] would yield linear OA(25⁸³, 975, F₂₅, 72) (dual of [975, 892, 73]-code), but
- the Johnson bound shows that NÂ â‰¤ 89837 389092 897810 364487 494581 967570 610542 625534 796349 698122 943887 036164 447658 836860 856960 180158 460562 485384 144465 583949 657372 034398 878743 622167 508877 481151 158132 769683 556711 341624 235788 344867 907990 124793 073516 840385 668094 477248 261727 877190 306298 068350 188446 459202 309418 376104 040802 444376 590725 557405 524277 866375 718932 223056 902249 535086 222317 724248 988556 813785 157797 332668 038898 601057 369724 505257 757700 557472 079722 459812 221283 954828 069105 602287 877970 025424 101387 116285 590696 306147 976007 500117 568403 501073 194553 840764 101810 614890 034215 547072 606559 007256 381001 080562 310897 382150 447022 907763 692636 492645 390223 106603 001300 132006 176136 703753 838197 132253 200914 969949 781923 431033 834465 675793 706398 942078 157290 042056 286850 799512 415782 364097 886534 768681 397093 986252 030873 668754 907910 460586 041138 204806 777982 352526 239898 684957 194323 728043 014367 252192 798579 855191 020324 450074 586599 345082 474611 653538 451107 411236 375764 508807 162118 452694 896483 937656 431020 792928 467073 704819 340536 775781 057614 583068 642064 920260 385798 417448 849093 546363 625865 334183 568140 377866 069543 114771 259206 002332 142559 981576 399208 113842 634539 531868 005754 272364 009310 490490 328696 225972 736658 590661 345891 524591 693637 764933 995552 895356 471183 743575 901868 699259 404826 452534 103976 986471 438509 867649 < 25⁸⁹² [i]

(83−72, 83, 975)-Net in Base 25 — Upper bound on s

There is no (11, 83, 976)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 108 998772 346726 132209 809864 709645 904113 612150 911857 078880 869384 548212 367802 024321 814834 431844 745817 425993 935916 071425 > 25⁸³ [i]

Best Known (83−72, 83, s)-Nets in Base 25

(83−72, 83, 126)-Net over F25 — Constructive and digital

(83−72, 83, 974)-Net over F25 — Upper bound on s (digital)

(83−72, 83, 975)-Net in Base 25 — Upper bound on s

(83−72, 83, 126)-Net over F₂₅ — Constructive and digital

(83−72, 83, 974)-Net over F₂₅ — Upper bound on s (digital)