Best Known (123−28, 123, s)-Nets in Base 8
(123−28, 123, 2341)-Net over F8 — Constructive and digital
Digital (95, 123, 2341)-net over F8, using
- 81 times duplication [i] based on digital (94, 122, 2341)-net over F8, using
- net defined by OOA [i] based on linear OOA(8122, 2341, F8, 28, 28) (dual of [(2341, 28), 65426, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(8122, 32774, F8, 28) (dual of [32774, 32652, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(8122, 32779, F8, 28) (dual of [32779, 32657, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [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(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, 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(27) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(8122, 32779, F8, 28) (dual of [32779, 32657, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(8122, 32774, F8, 28) (dual of [32774, 32652, 29]-code), using
- net defined by OOA [i] based on linear OOA(8122, 2341, F8, 28, 28) (dual of [(2341, 28), 65426, 29]-NRT-code), using
(123−28, 123, 26032)-Net over F8 — Digital
Digital (95, 123, 26032)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8123, 26032, F8, 28) (dual of [26032, 25909, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(8123, 32781, F8, 28) (dual of [32781, 32658, 29]-code), using
- construction XX applied to Ce(27) ⊂ Ce(25) ⊂ Ce(24) [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(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(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(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(27) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(8123, 32781, F8, 28) (dual of [32781, 32658, 29]-code), using
(123−28, 123, large)-Net in Base 8 — Upper bound on s
There is no (95, 123, large)-net in base 8, because
- 26 times m-reduction [i] would yield (95, 97, large)-net in base 8, but