Best Known (137−38, 137, s)-Nets in Base 8
(137−38, 137, 1026)-Net over F8 — Constructive and digital
Digital (99, 137, 1026)-net over F8, using
- 5 times m-reduction [i] based on digital (99, 142, 1026)-net over F8, using
- trace code for nets [i] based on digital (28, 71, 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, 71, 513)-net over F64, using
(137−38, 137, 4640)-Net over F8 — Digital
Digital (99, 137, 4640)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8137, 4640, F8, 38) (dual of [4640, 4503, 39]-code), using
- 536 step Varšamov–Edel lengthening with (ri) = (1, 16 times 0, 1, 84 times 0, 1, 190 times 0, 1, 242 times 0) [i] based on linear OA(8133, 4100, F8, 38) (dual of [4100, 3967, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(8133, 4096, F8, 38) (dual of [4096, 3963, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(8129, 4096, F8, 37) (dual of [4096, 3967, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(37) ⊂ Ce(36) [i] based on
- 536 step Varšamov–Edel lengthening with (ri) = (1, 16 times 0, 1, 84 times 0, 1, 190 times 0, 1, 242 times 0) [i] based on linear OA(8133, 4100, F8, 38) (dual of [4100, 3967, 39]-code), using
(137−38, 137, 3680189)-Net in Base 8 — Upper bound on s
There is no (99, 137, 3680190)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 5288 457050 045180 411758 775001 436683 131330 928731 159677 767036 665634 935989 404731 240382 476316 412757 763421 556015 765526 458934 610666 > 8137 [i]