Best Known (90−28, 90, s)-Nets in Base 5
(90−28, 90, 258)-Net over F5 — Constructive and digital
Digital (62, 90, 258)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 14, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (48, 76, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 38, 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, 38, 126)-net over F25, using
- digital (0, 14, 6)-net over F5, using
(90−28, 90, 634)-Net over F5 — Digital
Digital (62, 90, 634)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(590, 634, F5, 28) (dual of [634, 544, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(590, 635, F5, 28) (dual of [635, 545, 29]-code), using
- construction XX applied to Ce(27) ⊂ Ce(25) ⊂ Ce(23) [i] based on
- linear OA(587, 625, F5, 28) (dual of [625, 538, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(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(51, 8, F5, 1) (dual of [8, 7, 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(51, 2, F5, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(27) ⊂ Ce(25) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(590, 635, F5, 28) (dual of [635, 545, 29]-code), using
(90−28, 90, 47063)-Net in Base 5 — Upper bound on s
There is no (62, 90, 47064)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 807 917226 442637 750061 481028 177381 667707 781693 256815 634695 352705 > 590 [i]