Best Known (125−20, 125, s)-Nets in Base 8
(125−20, 125, 209717)-Net over F8 — Constructive and digital
Digital (105, 125, 209717)-net over F8, using
- 82 times duplication [i] based on digital (103, 123, 209717)-net over F8, using
- net defined by OOA [i] based on linear OOA(8123, 209717, F8, 20, 20) (dual of [(209717, 20), 4194217, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(8123, 2097170, F8, 20) (dual of [2097170, 2097047, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8123, 2097176, F8, 20) (dual of [2097176, 2097053, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(8120, 2097152, F8, 20) (dual of [2097152, 2097032, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(899, 2097152, F8, 17) (dual of [2097152, 2097053, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(83, 24, F8, 2) (dual of [24, 21, 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(19) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(8123, 2097176, F8, 20) (dual of [2097176, 2097053, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(8123, 2097170, F8, 20) (dual of [2097170, 2097047, 21]-code), using
- net defined by OOA [i] based on linear OOA(8123, 209717, F8, 20, 20) (dual of [(209717, 20), 4194217, 21]-NRT-code), using
(125−20, 125, 1796040)-Net over F8 — Digital
Digital (105, 125, 1796040)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8125, 1796040, F8, 20) (dual of [1796040, 1795915, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8125, 2097179, F8, 20) (dual of [2097179, 2097054, 21]-code), using
- construction XX applied to Ce(19) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- linear OA(8120, 2097152, F8, 20) (dual of [2097152, 2097032, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(899, 2097152, F8, 17) (dual of [2097152, 2097053, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(892, 2097152, F8, 15) (dual of [2097152, 2097060, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(83, 25, F8, 2) (dual of [25, 22, 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
- linear OA(81, 2, F8, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(19) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(8125, 2097179, F8, 20) (dual of [2097179, 2097054, 21]-code), using
(125−20, 125, large)-Net in Base 8 — Upper bound on s
There is no (105, 125, large)-net in base 8, because
- 18 times m-reduction [i] would yield (105, 107, large)-net in base 8, but