Best Known (118−18, 118, s)-Nets in Base 8
(118−18, 118, 233041)-Net over F8 — Constructive and digital
Digital (100, 118, 233041)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (3, 12, 24)-net over F8, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- the Klein quartic over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- digital (88, 106, 233017)-net over F8, using
- net defined by OOA [i] based on linear OOA(8106, 233017, F8, 18, 18) (dual of [(233017, 18), 4194200, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(8106, 2097153, F8, 18) (dual of [2097153, 2097047, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(8106, 2097159, F8, 18) (dual of [2097159, 2097053, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [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(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(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(8106, 2097159, F8, 18) (dual of [2097159, 2097053, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(8106, 2097153, F8, 18) (dual of [2097153, 2097047, 19]-code), using
- net defined by OOA [i] based on linear OOA(8106, 233017, F8, 18, 18) (dual of [(233017, 18), 4194200, 19]-NRT-code), using
- digital (3, 12, 24)-net over F8, using
(118−18, 118, 2097213)-Net over F8 — Digital
Digital (100, 118, 2097213)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8118, 2097213, F8, 18) (dual of [2097213, 2097095, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(9) [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(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(812, 61, F8, 7) (dual of [61, 49, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(812, 73, F8, 7) (dual of [73, 61, 8]-code), using
- a “DaH†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(812, 73, F8, 7) (dual of [73, 61, 8]-code), using
- construction X applied to Ce(17) ⊂ Ce(9) [i] based on
(118−18, 118, large)-Net in Base 8 — Upper bound on s
There is no (100, 118, large)-net in base 8, because
- 16 times m-reduction [i] would yield (100, 102, large)-net in base 8, but