Best Known (126, 126+37, s)-Nets in Base 8
(126, 126+37, 1821)-Net over F8 — Constructive and digital
Digital (126, 163, 1821)-net over F8, using
- 81 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
(126, 126+37, 30052)-Net over F8 — Digital
Digital (126, 163, 30052)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8163, 30052, F8, 37) (dual of [30052, 29889, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(8163, 32781, F8, 37) (dual of [32781, 32618, 38]-code), using
- 1 times code embedding in larger space [i] 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
- 1 times code embedding in larger space [i] based on linear OA(8162, 32780, F8, 37) (dual of [32780, 32618, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(8163, 32781, F8, 37) (dual of [32781, 32618, 38]-code), using
(126, 126+37, large)-Net in Base 8 — Upper bound on s
There is no (126, 163, large)-net in base 8, because
- 35 times m-reduction [i] would yield (126, 128, large)-net in base 8, but