Best Known (74, 74+25, s)-Nets in Base 7
(74, 74+25, 688)-Net over F7 — Constructive and digital
Digital (74, 99, 688)-net over F7, using
- 7 times m-reduction [i] based on digital (74, 106, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 53, 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, 53, 344)-net over F49, using
(74, 74+25, 5056)-Net over F7 — Digital
Digital (74, 99, 5056)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(799, 5056, F7, 25) (dual of [5056, 4957, 26]-code), using
- 249 step Varšamov–Edel lengthening with (ri) = (1, 248 times 0) [i] based on linear OA(798, 4806, F7, 25) (dual of [4806, 4708, 26]-code), using
- trace code [i] based on linear OA(4949, 2403, F49, 25) (dual of [2403, 2354, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(4949, 2401, F49, 25) (dual of [2401, 2352, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4947, 2401, F49, 24) (dual of [2401, 2354, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(490, 2, F49, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- trace code [i] based on linear OA(4949, 2403, F49, 25) (dual of [2403, 2354, 26]-code), using
- 249 step Varšamov–Edel lengthening with (ri) = (1, 248 times 0) [i] based on linear OA(798, 4806, F7, 25) (dual of [4806, 4708, 26]-code), using
(74, 74+25, 7028193)-Net in Base 7 — Upper bound on s
There is no (74, 99, 7028194)-net in base 7, because
- 1 times m-reduction [i] would yield (74, 98, 7028194)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 66009 811459 288234 876971 428231 196253 092156 575871 523637 691094 753204 083700 794419 174041 > 798 [i]