Best Known (57, 57+27, s)-Nets in Base 5
(57, 57+27, 252)-Net over F5 — Constructive and digital
Digital (57, 84, 252)-net over F5, using
- 10 times m-reduction [i] based on digital (57, 94, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 47, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 47, 126)-net over F25, using
(57, 57+27, 516)-Net over F5 — Digital
Digital (57, 84, 516)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(584, 516, F5, 27) (dual of [516, 432, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(584, 628, F5, 27) (dual of [628, 544, 28]-code), using
- 1 times code embedding in larger space [i] based on linear OA(583, 627, F5, 27) (dual of [627, 544, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(583, 625, F5, 27) (dual of [625, 542, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(581, 625, F5, 26) (dual of [625, 544, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(50, 2, F5, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(583, 627, F5, 27) (dual of [627, 544, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(584, 628, F5, 27) (dual of [628, 544, 28]-code), using
(57, 57+27, 41105)-Net in Base 5 — Upper bound on s
There is no (57, 84, 41106)-net in base 5, because
- 1 times m-reduction [i] would yield (57, 83, 41106)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 10340 406815 043377 418587 774847 550297 343756 607555 569354 289225 > 583 [i]