Best Known (59, 59+11, s)-Nets in Base 8
(59, 59+11, 419445)-Net over F8 — Constructive and digital
Digital (59, 70, 419445)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 14)-net over F8, using
- net from sequence [i] based on digital (1, 13)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 1 and N(F) ≥ 14, using
- net from sequence [i] based on digital (1, 13)-sequence over F8, using
- digital (53, 64, 419431)-net over F8, using
- net defined by OOA [i] based on linear OOA(864, 419431, F8, 11, 11) (dual of [(419431, 11), 4613677, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(864, 2097156, F8, 11) (dual of [2097156, 2097092, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(864, 2097159, F8, 11) (dual of [2097159, 2097095, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(864, 2097152, F8, 11) (dual of [2097152, 2097088, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(857, 2097152, F8, 10) (dual of [2097152, 2097095, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(80, 7, F8, 0) (dual of [7, 7, 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
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(864, 2097159, F8, 11) (dual of [2097159, 2097095, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(864, 2097156, F8, 11) (dual of [2097156, 2097092, 12]-code), using
- net defined by OOA [i] based on linear OOA(864, 419431, F8, 11, 11) (dual of [(419431, 11), 4613677, 12]-NRT-code), using
- digital (1, 6, 14)-net over F8, using
(59, 59+11, 2097186)-Net over F8 — Digital
Digital (59, 70, 2097186)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(870, 2097186, F8, 11) (dual of [2097186, 2097116, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
- linear OA(864, 2097152, F8, 11) (dual of [2097152, 2097088, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(836, 2097152, F8, 6) (dual of [2097152, 2097116, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(86, 34, F8, 4) (dual of [34, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- 1 times truncation [i] based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
(59, 59+11, large)-Net in Base 8 — Upper bound on s
There is no (59, 70, large)-net in base 8, because
- 9 times m-reduction [i] would yield (59, 61, large)-net in base 8, but