Best Known (88, 88+20, s)-Nets in Base 8
(88, 88+20, 26216)-Net over F8 — Constructive and digital
Digital (88, 108, 26216)-net over F8, using
- 82 times duplication [i] based on digital (86, 106, 26216)-net over F8, using
- net defined by OOA [i] based on linear OOA(8106, 26216, F8, 20, 20) (dual of [(26216, 20), 524214, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(8106, 262160, F8, 20) (dual of [262160, 262054, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8106, 262165, F8, 20) (dual of [262165, 262059, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(8103, 262144, F8, 20) (dual of [262144, 262041, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(885, 262144, F8, 17) (dual of [262144, 262059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(83, 21, F8, 2) (dual of [21, 18, 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
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(8106, 262165, F8, 20) (dual of [262165, 262059, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(8106, 262160, F8, 20) (dual of [262160, 262054, 21]-code), using
- net defined by OOA [i] based on linear OOA(8106, 26216, F8, 20, 20) (dual of [(26216, 20), 524214, 21]-NRT-code), using
(88, 88+20, 251989)-Net over F8 — Digital
Digital (88, 108, 251989)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8108, 251989, F8, 20) (dual of [251989, 251881, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8108, 262168, F8, 20) (dual of [262168, 262060, 21]-code), using
- construction XX applied to Ce(19) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- linear OA(8103, 262144, F8, 20) (dual of [262144, 262041, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(885, 262144, F8, 17) (dual of [262144, 262059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(879, 262144, F8, 15) (dual of [262144, 262065, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(83, 22, F8, 2) (dual of [22, 19, 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(19) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(8108, 262168, F8, 20) (dual of [262168, 262060, 21]-code), using
(88, 88+20, large)-Net in Base 8 — Upper bound on s
There is no (88, 108, large)-net in base 8, because
- 18 times m-reduction [i] would yield (88, 90, large)-net in base 8, but