Best Known (167−37, 167, s)-Nets in Base 8
(167−37, 167, 1821)-Net over F8 — Constructive and digital
Digital (130, 167, 1821)-net over F8, using
- 85 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
(167−37, 167, 32797)-Net over F8 — Digital
Digital (130, 167, 32797)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8167, 32797, F8, 37) (dual of [32797, 32630, 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, 32795, F8, 36) (dual of [32795, 32630, 37]-code), using Gilbert–Varšamov bound and bm = 8165 > Vbs−1(k−1) = 4066 177743 899096 602683 464230 709518 922722 918364 629672 474993 222635 458244 035026 712435 827672 893229 011831 204599 818743 978020 956346 979582 723990 639619 788800 [i]
- linear OA(80, 2, F8, 0) (dual of [2, 2, 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
(167−37, 167, large)-Net in Base 8 — Upper bound on s
There is no (130, 167, large)-net in base 8, because
- 35 times m-reduction [i] would yield (130, 132, large)-net in base 8, but