Best Known (57, 57+26, s)-Nets in Base 5
(57, 57+26, 252)-Net over F5 — Constructive and digital
Digital (57, 83, 252)-net over F5, using
- 11 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+26, 583)-Net over F5 — Digital
Digital (57, 83, 583)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(583, 583, F5, 26) (dual of [583, 500, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(583, 632, F5, 26) (dual of [632, 549, 27]-code), using
- construction XX applied to Ce(25) ⊂ Ce(23) ⊂ Ce(22) [i] based on
- 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(577, 625, F5, 24) (dual of [625, 548, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(573, 625, F5, 23) (dual of [625, 552, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(51, 6, F5, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(25) ⊂ Ce(23) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(583, 632, F5, 26) (dual of [632, 549, 27]-code), using
(57, 57+26, 41105)-Net in Base 5 — Upper bound on s
There is no (57, 83, 41106)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 10340 406815 043377 418587 774847 550297 343756 607555 569354 289225 > 583 [i]