Best Known (73, 73+29, s)-Nets in Base 7
(73, 73+29, 688)-Net over F7 — Constructive and digital
Digital (73, 102, 688)-net over F7, using
- 2 times m-reduction [i] based on digital (73, 104, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 52, 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, 52, 344)-net over F49, using
(73, 73+29, 2454)-Net over F7 — Digital
Digital (73, 102, 2454)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7102, 2454, F7, 29) (dual of [2454, 2352, 30]-code), using
- 44 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 10 times 0, 1, 28 times 0) [i] based on linear OA(798, 2406, F7, 29) (dual of [2406, 2308, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- 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(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(71, 5, F7, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- 44 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 10 times 0, 1, 28 times 0) [i] based on linear OA(798, 2406, F7, 29) (dual of [2406, 2308, 30]-code), using
(73, 73+29, 1259163)-Net in Base 7 — Upper bound on s
There is no (73, 102, 1259164)-net in base 7, because
- 1 times m-reduction [i] would yield (73, 101, 1259164)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 22 641436 950734 214681 680177 575169 749854 733823 218645 451736 414327 680652 894978 583419 211081 > 7101 [i]