Best Known (72, 72+24, s)-Nets in Base 7
(72, 72+24, 688)-Net over F7 — Constructive and digital
Digital (72, 96, 688)-net over F7, using
- 6 times m-reduction [i] based on digital (72, 102, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 51, 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, 51, 344)-net over F49, using
(72, 72+24, 5322)-Net over F7 — Digital
Digital (72, 96, 5322)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(796, 5322, F7, 24) (dual of [5322, 5226, 25]-code), using
- 514 step Varšamov–Edel lengthening with (ri) = (1, 164 times 0, 1, 348 times 0) [i] based on linear OA(794, 4806, F7, 24) (dual of [4806, 4712, 25]-code), using
- trace code [i] based on linear OA(4947, 2403, F49, 24) (dual of [2403, 2356, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- 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(4945, 2401, F49, 23) (dual of [2401, 2356, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(23) ⊂ Ce(22) [i] based on
- trace code [i] based on linear OA(4947, 2403, F49, 24) (dual of [2403, 2356, 25]-code), using
- 514 step Varšamov–Edel lengthening with (ri) = (1, 164 times 0, 1, 348 times 0) [i] based on linear OA(794, 4806, F7, 24) (dual of [4806, 4712, 25]-code), using
(72, 72+24, 5081522)-Net in Base 7 — Upper bound on s
There is no (72, 96, 5081523)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 1347 139219 105247 455618 999118 065466 123975 001210 243783 039741 037541 449152 832643 836633 > 796 [i]