Best Known (57−13, 57, s)-Nets in Base 5
(57−13, 57, 527)-Net over F5 — Constructive and digital
Digital (44, 57, 527)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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 (38, 51, 521)-net over F5, using
- net defined by OOA [i] based on linear OOA(551, 521, F5, 13, 13) (dual of [(521, 13), 6722, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(551, 3127, F5, 13) (dual of [3127, 3076, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(551, 3130, F5, 13) (dual of [3130, 3079, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(551, 3125, F5, 13) (dual of [3125, 3074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 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(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(551, 3130, F5, 13) (dual of [3130, 3079, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(551, 3127, F5, 13) (dual of [3127, 3076, 14]-code), using
- net defined by OOA [i] based on linear OOA(551, 521, F5, 13, 13) (dual of [(521, 13), 6722, 14]-NRT-code), using
- digital (0, 6, 6)-net over F5, using
(57−13, 57, 3296)-Net over F5 — Digital
Digital (44, 57, 3296)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(557, 3296, F5, 13) (dual of [3296, 3239, 14]-code), using
- 160 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0, 1, 18 times 0, 1, 41 times 0, 1, 87 times 0) [i] based on linear OA(551, 3130, F5, 13) (dual of [3130, 3079, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(551, 3125, F5, 13) (dual of [3125, 3074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 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(12) ⊂ Ce(11) [i] based on
- 160 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0, 1, 18 times 0, 1, 41 times 0, 1, 87 times 0) [i] based on linear OA(551, 3130, F5, 13) (dual of [3130, 3079, 14]-code), using
(57−13, 57, 2499662)-Net in Base 5 — Upper bound on s
There is no (44, 57, 2499663)-net in base 5, because
- 1 times m-reduction [i] would yield (44, 56, 2499663)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 1387 779258 445059 084284 969492 280846 884329 > 556 [i]