Best Known (92−25, 92, s)-Nets in Base 7
(92−25, 92, 688)-Net over F7 — Constructive and digital
Digital (67, 92, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 46, 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
(92−25, 92, 2859)-Net over F7 — Digital
Digital (67, 92, 2859)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(792, 2859, F7, 25) (dual of [2859, 2767, 26]-code), using
- 447 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 31 times 0, 1, 72 times 0, 1, 133 times 0, 1, 191 times 0) [i] based on linear OA(785, 2405, F7, 25) (dual of [2405, 2320, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(785, 2401, F7, 25) (dual of [2401, 2316, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(781, 2401, F7, 24) (dual of [2401, 2320, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(24) ⊂ Ce(23) [i] based on
- 447 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 31 times 0, 1, 72 times 0, 1, 133 times 0, 1, 191 times 0) [i] based on linear OA(785, 2405, F7, 25) (dual of [2405, 2320, 26]-code), using
(92−25, 92, 2258753)-Net in Base 7 — Upper bound on s
There is no (67, 92, 2258754)-net in base 7, because
- 1 times m-reduction [i] would yield (67, 91, 2258754)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 80153 512864 550438 165867 350318 392467 641877 922489 201979 055287 328570 563147 330201 > 791 [i]