Best Known (33, 41, s)-Nets in Base 5
(33, 41, 3913)-Net over F5 — Constructive and digital
Digital (33, 41, 3913)-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 (29, 37, 3907)-net over F5, using
- net defined by OOA [i] based on linear OOA(537, 3907, F5, 8, 8) (dual of [(3907, 8), 31219, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(537, 15628, F5, 8) (dual of [15628, 15591, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(537, 15631, F5, 8) (dual of [15631, 15594, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(537, 15625, F5, 8) (dual of [15625, 15588, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(537, 15631, F5, 8) (dual of [15631, 15594, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(537, 15628, F5, 8) (dual of [15628, 15591, 9]-code), using
- net defined by OOA [i] based on linear OOA(537, 3907, F5, 8, 8) (dual of [(3907, 8), 31219, 9]-NRT-code), using
- digital (0, 4, 6)-net over F5, using
(33, 41, 15647)-Net over F5 — Digital
Digital (33, 41, 15647)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(541, 15647, F5, 8) (dual of [15647, 15606, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
- linear OA(537, 15625, F5, 8) (dual of [15625, 15588, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(54, 22, F5, 3) (dual of [22, 18, 4]-code or 22-cap in PG(3,5)), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
(33, 41, 8080444)-Net in Base 5 — Upper bound on s
There is no (33, 41, 8080445)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 45474 755772 445667 093761 775601 > 541 [i]