Best Known (75, 75+30, s)-Nets in Base 7
(75, 75+30, 688)-Net over F7 — Constructive and digital
Digital (75, 105, 688)-net over F7, using
- 3 times m-reduction [i] based on digital (75, 108, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 54, 344)-net over F49, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- the Hermitian function field over F49 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- trace code for nets [i] based on digital (21, 54, 344)-net over F49, using
(75, 75+30, 2445)-Net over F7 — Digital
Digital (75, 105, 2445)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7105, 2445, F7, 30) (dual of [2445, 2340, 31]-code), using
- 36 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 8 times 0, 1, 23 times 0) [i] based on linear OA(7101, 2405, F7, 30) (dual of [2405, 2304, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- linear OA(7101, 2401, F7, 30) (dual of [2401, 2300, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(797, 2401, F7, 29) (dual of [2401, 2304, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- 36 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 8 times 0, 1, 23 times 0) [i] based on linear OA(7101, 2405, F7, 30) (dual of [2405, 2304, 31]-code), using
(75, 75+30, 881651)-Net in Base 7 — Upper bound on s
There is no (75, 105, 881652)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 54362 233773 227255 760164 488028 641085 189223 455238 802317 617891 256866 500738 521373 466709 097633 > 7105 [i]