Best Known (127, 127+37, s)-Nets in Base 8
(127, 127+37, 1821)-Net over F8 — Constructive and digital
Digital (127, 164, 1821)-net over F8, using
- 82 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
(127, 127+37, 31893)-Net over F8 — Digital
Digital (127, 164, 31893)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8164, 31893, F8, 37) (dual of [31893, 31729, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(8164, 32786, F8, 37) (dual of [32786, 32622, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- linear OA(8161, 32768, F8, 37) (dual of [32768, 32607, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(8146, 32768, F8, 34) (dual of [32768, 32622, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(83, 18, F8, 2) (dual of [18, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(8164, 32786, F8, 37) (dual of [32786, 32622, 38]-code), using
(127, 127+37, large)-Net in Base 8 — Upper bound on s
There is no (127, 164, large)-net in base 8, because
- 35 times m-reduction [i] would yield (127, 129, large)-net in base 8, but