Best Known (107, 107+42, s)-Nets in Base 8
(107, 107+42, 1026)-Net over F8 — Constructive and digital
Digital (107, 149, 1026)-net over F8, using
- 9 times m-reduction [i] based on digital (107, 158, 1026)-net over F8, using
- trace code for nets [i] based on digital (28, 79, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 79, 513)-net over F64, using
(107, 107+42, 4438)-Net over F8 — Digital
Digital (107, 149, 4438)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8149, 4438, F8, 42) (dual of [4438, 4289, 43]-code), using
- 334 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 27 times 0, 1, 108 times 0, 1, 192 times 0) [i] based on linear OA(8145, 4100, F8, 42) (dual of [4100, 3955, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(40) [i] based on
- linear OA(8145, 4096, F8, 42) (dual of [4096, 3951, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(8141, 4096, F8, 41) (dual of [4096, 3955, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 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(41) ⊂ Ce(40) [i] based on
- 334 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 27 times 0, 1, 108 times 0, 1, 192 times 0) [i] based on linear OA(8145, 4100, F8, 42) (dual of [4100, 3955, 43]-code), using
(107, 107+42, 3169795)-Net in Base 8 — Upper bound on s
There is no (107, 149, 3169796)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 363 420905 374023 500615 399529 329093 620455 972575 026703 111288 998116 856239 869416 103210 401185 819811 118131 029915 654776 526326 114279 685616 695168 > 8149 [i]