Best Known (128−28, 128, s)-Nets in Base 8
(128−28, 128, 2342)-Net over F8 — Constructive and digital
Digital (100, 128, 2342)-net over F8, using
- 82 times duplication [i] based on digital (98, 126, 2342)-net over F8, using
- net defined by OOA [i] based on linear OOA(8126, 2342, F8, 28, 28) (dual of [(2342, 28), 65450, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(8126, 32788, F8, 28) (dual of [32788, 32662, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(8126, 32789, F8, 28) (dual of [32789, 32663, 29]-code), using
- construction XX applied to Ce(27) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(8121, 32768, F8, 28) (dual of [32768, 32647, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(8106, 32768, F8, 25) (dual of [32768, 32662, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(8101, 32768, F8, 23) (dual of [32768, 32667, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(83, 19, F8, 2) (dual of [19, 16, 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(27) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(8126, 32789, F8, 28) (dual of [32789, 32663, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(8126, 32788, F8, 28) (dual of [32788, 32662, 29]-code), using
- net defined by OOA [i] based on linear OOA(8126, 2342, F8, 28, 28) (dual of [(2342, 28), 65450, 29]-NRT-code), using
(128−28, 128, 32800)-Net over F8 — Digital
Digital (100, 128, 32800)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8128, 32800, F8, 28) (dual of [32800, 32672, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- linear OA(8121, 32768, F8, 28) (dual of [32768, 32647, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(896, 32768, F8, 22) (dual of [32768, 32672, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(87, 32, F8, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
(128−28, 128, large)-Net in Base 8 — Upper bound on s
There is no (100, 128, large)-net in base 8, because
- 26 times m-reduction [i] would yield (100, 102, large)-net in base 8, but