Best Known (90, 90+26, s)-Nets in Base 8
(90, 90+26, 2522)-Net over F8 — Constructive and digital
Digital (90, 116, 2522)-net over F8, using
- 81 times duplication [i] based on digital (89, 115, 2522)-net over F8, using
- net defined by OOA [i] based on linear OOA(8115, 2522, F8, 26, 26) (dual of [(2522, 26), 65457, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(8115, 32786, F8, 26) (dual of [32786, 32671, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8115, 32787, F8, 26) (dual of [32787, 32672, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- 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(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(84, 19, F8, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,8)), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(8115, 32787, F8, 26) (dual of [32787, 32672, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(8115, 32786, F8, 26) (dual of [32786, 32671, 27]-code), using
- net defined by OOA [i] based on linear OOA(8115, 2522, F8, 26, 26) (dual of [(2522, 26), 65457, 27]-NRT-code), using
(90, 90+26, 29741)-Net over F8 — Digital
Digital (90, 116, 29741)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8116, 29741, F8, 26) (dual of [29741, 29625, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8116, 32789, F8, 26) (dual of [32789, 32673, 27]-code), using
- construction XX applied to Ce(25) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- 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(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(891, 32768, F8, 21) (dual of [32768, 32677, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(84, 20, F8, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,8)), 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(25) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(8116, 32789, F8, 26) (dual of [32789, 32673, 27]-code), using
(90, 90+26, large)-Net in Base 8 — Upper bound on s
There is no (90, 116, large)-net in base 8, because
- 24 times m-reduction [i] would yield (90, 92, large)-net in base 8, but