Best Known (114−31, 114, s)-Nets in Base 8
(114−31, 114, 562)-Net over F8 — Constructive and digital
Digital (83, 114, 562)-net over F8, using
- 82 times duplication [i] based on digital (81, 112, 562)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (21, 36, 208)-net over F8, using
- trace code for nets [i] based on digital (3, 18, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 18, 104)-net over F64, using
- digital (45, 76, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 38, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 38, 177)-net over F64, using
- digital (21, 36, 208)-net over F8, using
- (u, u+v)-construction [i] based on
(114−31, 114, 654)-Net in Base 8 — Constructive
(83, 114, 654)-net in base 8, using
- (u, u+v)-construction [i] based on
- (23, 38, 300)-net in base 8, using
- trace code for nets [i] based on (4, 19, 150)-net in base 64, using
- 2 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
- base change [i] based on digital (1, 18, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- base change [i] based on digital (1, 18, 150)-net over F128, using
- 2 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
- trace code for nets [i] based on (4, 19, 150)-net in base 64, using
- digital (45, 76, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 38, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 38, 177)-net over F64, using
- (23, 38, 300)-net in base 8, using
(114−31, 114, 4668)-Net over F8 — Digital
Digital (83, 114, 4668)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8114, 4668, F8, 31) (dual of [4668, 4554, 32]-code), using
- 559 step Varšamov–Edel lengthening with (ri) = (1, 9 times 0, 1, 62 times 0, 1, 194 times 0, 1, 290 times 0) [i] based on linear OA(8110, 4105, F8, 31) (dual of [4105, 3995, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(8109, 4096, F8, 31) (dual of [4096, 3987, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(8101, 4096, F8, 29) (dual of [4096, 3995, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(81, 9, F8, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- 559 step Varšamov–Edel lengthening with (ri) = (1, 9 times 0, 1, 62 times 0, 1, 194 times 0, 1, 290 times 0) [i] based on linear OA(8110, 4105, F8, 31) (dual of [4105, 3995, 32]-code), using
(114−31, 114, 5833723)-Net in Base 8 — Upper bound on s
There is no (83, 114, 5833724)-net in base 8, because
- 1 times m-reduction [i] would yield (83, 113, 5833724)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 1 119874 545818 995299 772016 499830 573334 456965 450947 475319 356692 330840 414511 185152 246966 409105 697957 505956 > 8113 [i]