Best Known (70, 99, s)-Nets in Base 7
(70, 99, 344)-Net over F7 — Constructive and digital
Digital (70, 99, 344)-net over F7, using
- base reduction for projective spaces (embedding PG(49,49) in PG(98,7)) for nets [i] based on digital (21, 50, 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
(70, 99, 2110)-Net over F7 — Digital
Digital (70, 99, 2110)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(799, 2110, F7, 29) (dual of [2110, 2011, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(799, 2409, F7, 29) (dual of [2409, 2310, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [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(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(72, 8, F7, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,7)), using
- extended Reed–Solomon code RSe(6,7) [i]
- Hamming code H(2,7) [i]
- algebraic-geometric code AG(F, Q+1P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using the rational function field F7(x) [i]
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(799, 2409, F7, 29) (dual of [2409, 2310, 30]-code), using
(70, 99, 829828)-Net in Base 7 — Upper bound on s
There is no (70, 99, 829829)-net in base 7, because
- 1 times m-reduction [i] would yield (70, 98, 829829)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 66010 707361 132372 842196 612395 896340 482195 023987 928621 907279 177591 778555 682137 278553 > 798 [i]