Best Known (128−29, 128, s)-Nets in Base 8
(128−29, 128, 2341)-Net over F8 — Constructive and digital
Digital (99, 128, 2341)-net over F8, using
- 81 times duplication [i] based on digital (98, 127, 2341)-net over F8, using
- net defined by OOA [i] based on linear OOA(8127, 2341, F8, 29, 29) (dual of [(2341, 29), 67762, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(8127, 32775, F8, 29) (dual of [32775, 32648, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(8127, 32779, F8, 29) (dual of [32779, 32652, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(8126, 32768, F8, 29) (dual of [32768, 32642, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(8116, 32768, F8, 27) (dual of [32768, 32652, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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 Ce(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(8127, 32779, F8, 29) (dual of [32779, 32652, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(8127, 32775, F8, 29) (dual of [32775, 32648, 30]-code), using
- net defined by OOA [i] based on linear OOA(8127, 2341, F8, 29, 29) (dual of [(2341, 29), 67762, 30]-NRT-code), using
(128−29, 128, 27601)-Net over F8 — Digital
Digital (99, 128, 27601)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8128, 27601, F8, 29) (dual of [27601, 27473, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(8128, 32781, F8, 29) (dual of [32781, 32653, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- linear OA(8126, 32768, F8, 29) (dual of [32768, 32642, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(8116, 32768, F8, 27) (dual of [32768, 32652, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8111, 32768, F8, 26) (dual of [32768, 32657, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(81, 12, F8, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(28) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(8128, 32781, F8, 29) (dual of [32781, 32653, 30]-code), using
(128−29, 128, large)-Net in Base 8 — Upper bound on s
There is no (99, 128, large)-net in base 8, because
- 27 times m-reduction [i] would yield (99, 101, large)-net in base 8, but