Best Known (129, 129+37, s)-Nets in Base 8
(129, 129+37, 1821)-Net over F8 — Constructive and digital
Digital (129, 166, 1821)-net over F8, using
- 84 times duplication [i] based on digital (125, 162, 1821)-net over F8, using
- net defined by OOA [i] based on linear OOA(8162, 1821, F8, 37, 37) (dual of [(1821, 37), 67215, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(8162, 32779, F8, 37) (dual of [32779, 32617, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(8162, 32780, F8, 37) (dual of [32780, 32618, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,17]) [i] based on
- linear OA(8161, 32769, F8, 37) (dual of [32769, 32608, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(8151, 32769, F8, 35) (dual of [32769, 32618, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,18]) ⊂ C([0,17]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8162, 32780, F8, 37) (dual of [32780, 32618, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(8162, 32779, F8, 37) (dual of [32779, 32617, 38]-code), using
- net defined by OOA [i] based on linear OOA(8162, 1821, F8, 37, 37) (dual of [(1821, 37), 67215, 38]-NRT-code), using
(129, 129+37, 32795)-Net over F8 — Digital
Digital (129, 166, 32795)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8166, 32795, F8, 37) (dual of [32795, 32629, 38]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8165, 32793, F8, 37) (dual of [32793, 32628, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(8161, 32769, F8, 37) (dual of [32769, 32608, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(8141, 32769, F8, 33) (dual of [32769, 32628, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(84, 24, F8, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,8)), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(8165, 32794, F8, 36) (dual of [32794, 32629, 37]-code), using Gilbert–Varšamov bound and bm = 8165 > Vbs−1(k−1) = 4061 838059 799988 339067 713943 399822 023728 479878 669542 849993 209157 480319 613172 275294 669877 887824 353486 853253 596991 962957 231732 850158 110912 541079 044096 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(8165, 32793, F8, 37) (dual of [32793, 32628, 38]-code), using
- construction X with Varšamov bound [i] based on
(129, 129+37, large)-Net in Base 8 — Upper bound on s
There is no (129, 166, large)-net in base 8, because
- 35 times m-reduction [i] would yield (129, 131, large)-net in base 8, but