Best Known (168−38, 168, s)-Nets in Base 8
(168−38, 168, 1725)-Net over F8 — Constructive and digital
Digital (130, 168, 1725)-net over F8, using
- 81 times duplication [i] based on digital (129, 167, 1725)-net over F8, using
- net defined by OOA [i] based on linear OOA(8167, 1725, F8, 38, 38) (dual of [(1725, 38), 65383, 39]-NRT-code), using
- OA 19-folding and stacking [i] based on linear OA(8167, 32775, F8, 38) (dual of [32775, 32608, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(8167, 32779, F8, 38) (dual of [32779, 32612, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(35) [i] based on
- linear OA(8166, 32768, F8, 38) (dual of [32768, 32602, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(8156, 32768, F8, 36) (dual of [32768, 32612, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [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(37) ⊂ Ce(35) [i] based on
- discarding factors / shortening the dual code based on linear OA(8167, 32779, F8, 38) (dual of [32779, 32612, 39]-code), using
- OA 19-folding and stacking [i] based on linear OA(8167, 32775, F8, 38) (dual of [32775, 32608, 39]-code), using
- net defined by OOA [i] based on linear OOA(8167, 1725, F8, 38, 38) (dual of [(1725, 38), 65383, 39]-NRT-code), using
(168−38, 168, 31526)-Net over F8 — Digital
Digital (130, 168, 31526)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8168, 31526, F8, 38) (dual of [31526, 31358, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(8168, 32781, F8, 38) (dual of [32781, 32613, 39]-code), using
- construction XX applied to Ce(37) ⊂ Ce(35) ⊂ Ce(34) [i] based on
- linear OA(8166, 32768, F8, 38) (dual of [32768, 32602, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(8156, 32768, F8, 36) (dual of [32768, 32612, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(8151, 32768, F8, 35) (dual of [32768, 32617, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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(37) ⊂ Ce(35) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(8168, 32781, F8, 38) (dual of [32781, 32613, 39]-code), using
(168−38, 168, large)-Net in Base 8 — Upper bound on s
There is no (130, 168, large)-net in base 8, because
- 36 times m-reduction [i] would yield (130, 132, large)-net in base 8, but