Best Known (29−7, 29, s)-Nets in Base 5
(29−7, 29, 1049)-Net over F5 — Constructive and digital
Digital (22, 29, 1049)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 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 (19, 26, 1043)-net over F5, using
- net defined by OOA [i] based on linear OOA(526, 1043, F5, 7, 7) (dual of [(1043, 7), 7275, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(526, 3130, F5, 7) (dual of [3130, 3104, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(526, 3125, F5, 7) (dual of [3125, 3099, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(521, 3125, F5, 6) (dual of [3125, 3104, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(526, 3130, F5, 7) (dual of [3130, 3104, 8]-code), using
- net defined by OOA [i] based on linear OOA(526, 1043, F5, 7, 7) (dual of [(1043, 7), 7275, 8]-NRT-code), using
- digital (0, 3, 6)-net over F5, using
(29−7, 29, 3160)-Net over F5 — Digital
Digital (22, 29, 3160)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(529, 3160, F5, 7) (dual of [3160, 3131, 8]-code), using
- 27 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 22 times 0) [i] based on linear OA(526, 3130, F5, 7) (dual of [3130, 3104, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(526, 3125, F5, 7) (dual of [3125, 3099, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(521, 3125, F5, 6) (dual of [3125, 3104, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- 27 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 22 times 0) [i] based on linear OA(526, 3130, F5, 7) (dual of [3130, 3104, 8]-code), using
(29−7, 29, 1517201)-Net in Base 5 — Upper bound on s
There is no (22, 29, 1517202)-net in base 5, because
- 1 times m-reduction [i] would yield (22, 28, 1517202)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 37 252930 824900 507625 > 528 [i]