Best Known (27, 35, s)-Nets in Base 5
(27, 35, 788)-Net over F5 — Constructive and digital
Digital (27, 35, 788)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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 (23, 31, 782)-net over F5, using
- net defined by OOA [i] based on linear OOA(531, 782, F5, 8, 8) (dual of [(782, 8), 6225, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(531, 3128, F5, 8) (dual of [3128, 3097, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(531, 3130, F5, 8) (dual of [3130, 3099, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(531, 3125, F5, 8) (dual of [3125, 3094, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(531, 3130, F5, 8) (dual of [3130, 3099, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(531, 3128, F5, 8) (dual of [3128, 3097, 9]-code), using
- net defined by OOA [i] based on linear OOA(531, 782, F5, 8, 8) (dual of [(782, 8), 6225, 9]-NRT-code), using
- digital (0, 4, 6)-net over F5, using
(27, 35, 3266)-Net over F5 — Digital
Digital (27, 35, 3266)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(535, 3266, F5, 8) (dual of [3266, 3231, 9]-code), using
- 132 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 23 times 0, 1, 103 times 0) [i] based on linear OA(531, 3130, F5, 8) (dual of [3130, 3099, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(531, 3125, F5, 8) (dual of [3125, 3094, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(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(7) ⊂ Ce(6) [i] based on
- 132 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 23 times 0, 1, 103 times 0) [i] based on linear OA(531, 3130, F5, 8) (dual of [3130, 3099, 9]-code), using
(27, 35, 722734)-Net in Base 5 — Upper bound on s
There is no (27, 35, 722735)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 2 910389 266974 848174 562801 > 535 [i]