Best Known (108−18, 108, s)-Nets in Base 8
(108−18, 108, 233018)-Net over F8 — Constructive and digital
Digital (90, 108, 233018)-net over F8, using
- net defined by OOA [i] based on linear OOA(8108, 233018, F8, 18, 18) (dual of [(233018, 18), 4194216, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(8108, 2097162, F8, 18) (dual of [2097162, 2097054, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- linear OA(8106, 2097152, F8, 18) (dual of [2097152, 2097046, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(899, 2097152, F8, 17) (dual of [2097152, 2097053, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(892, 2097152, F8, 15) (dual of [2097152, 2097060, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(80, 8, F8, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, 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(17) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- OA 9-folding and stacking [i] based on linear OA(8108, 2097162, F8, 18) (dual of [2097162, 2097054, 19]-code), using
(108−18, 108, 1063778)-Net over F8 — Digital
Digital (90, 108, 1063778)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8108, 1063778, F8, 18) (dual of [1063778, 1063670, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(8108, 2097162, F8, 18) (dual of [2097162, 2097054, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- linear OA(8106, 2097152, F8, 18) (dual of [2097152, 2097046, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(899, 2097152, F8, 17) (dual of [2097152, 2097053, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(892, 2097152, F8, 15) (dual of [2097152, 2097060, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(80, 8, F8, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, 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(17) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(8108, 2097162, F8, 18) (dual of [2097162, 2097054, 19]-code), using
(108−18, 108, large)-Net in Base 8 — Upper bound on s
There is no (90, 108, large)-net in base 8, because
- 16 times m-reduction [i] would yield (90, 92, large)-net in base 8, but