Best Known (72, 99, s)-Nets in Base 7
(72, 99, 688)-Net over F7 — Constructive and digital
Digital (72, 99, 688)-net over F7, using
- 3 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, 99, 2919)-Net over F7 — Digital
Digital (72, 99, 2919)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(799, 2919, F7, 27) (dual of [2919, 2820, 28]-code), using
- 508 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 33 times 0, 1, 100 times 0, 1, 165 times 0, 1, 200 times 0) [i] based on linear OA(793, 2405, F7, 27) (dual of [2405, 2312, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(793, 2401, F7, 27) (dual of [2401, 2308, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(789, 2401, F7, 26) (dual of [2401, 2312, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(26) ⊂ Ce(25) [i] based on
- 508 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 33 times 0, 1, 100 times 0, 1, 165 times 0, 1, 200 times 0) [i] based on linear OA(793, 2405, F7, 27) (dual of [2405, 2312, 28]-code), using
(72, 99, 2218157)-Net in Base 7 — Upper bound on s
There is no (72, 99, 2218158)-net in base 7, because
- 1 times m-reduction [i] would yield (72, 98, 2218158)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 66009 879013 314454 303045 401903 244191 870908 104390 776119 809602 420253 980241 830458 851069 > 798 [i]