Best Known (27, 34, s)-Nets in Base 5
(27, 34, 5216)-Net over F5 — Constructive and digital
Digital (27, 34, 5216)-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 (24, 31, 5210)-net over F5, using
- net defined by OOA [i] based on linear OOA(531, 5210, F5, 7, 7) (dual of [(5210, 7), 36439, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(531, 15631, F5, 7) (dual of [15631, 15600, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(531, 15625, F5, 7) (dual of [15625, 15594, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(525, 15625, F5, 6) (dual of [15625, 15600, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(50, 6, F5, 0) (dual of [6, 6, 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(531, 15631, F5, 7) (dual of [15631, 15600, 8]-code), using
- net defined by OOA [i] based on linear OOA(531, 5210, F5, 7, 7) (dual of [(5210, 7), 36439, 8]-NRT-code), using
- digital (0, 3, 6)-net over F5, using
(27, 34, 15640)-Net over F5 — Digital
Digital (27, 34, 15640)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(534, 15640, F5, 7) (dual of [15640, 15606, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(531, 15625, F5, 7) (dual of [15625, 15594, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(519, 15625, F5, 4) (dual of [15625, 15606, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(53, 15, F5, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
(27, 34, large)-Net in Base 5 — Upper bound on s
There is no (27, 34, large)-net in base 5, because
- 5 times m-reduction [i] would yield (27, 29, large)-net in base 5, but