Best Known (40, 40+12, s)-Nets in Base 5
(40, 40+12, 527)-Net over F5 — Constructive and digital
Digital (40, 52, 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 (34, 46, 521)-net over F5, using
- net defined by OOA [i] based on linear OOA(546, 521, F5, 12, 12) (dual of [(521, 12), 6206, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(546, 3126, F5, 12) (dual of [3126, 3080, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(546, 3130, F5, 12) (dual of [3130, 3084, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- 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(541, 3125, F5, 11) (dual of [3125, 3084, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(546, 3130, F5, 12) (dual of [3130, 3084, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(546, 3126, F5, 12) (dual of [3126, 3080, 13]-code), using
- net defined by OOA [i] based on linear OOA(546, 521, F5, 12, 12) (dual of [(521, 12), 6206, 13]-NRT-code), using
- digital (0, 6, 6)-net over F5, using
(40, 40+12, 3240)-Net over F5 — Digital
Digital (40, 52, 3240)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(552, 3240, F5, 12) (dual of [3240, 3188, 13]-code), using
- 104 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 11 times 0, 1, 26 times 0, 1, 57 times 0) [i] based on linear OA(546, 3130, F5, 12) (dual of [3130, 3084, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- 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(541, 3125, F5, 11) (dual of [3125, 3084, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- 104 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 11 times 0, 1, 26 times 0, 1, 57 times 0) [i] based on linear OA(546, 3130, F5, 12) (dual of [3130, 3084, 13]-code), using
(40, 40+12, 854870)-Net in Base 5 — Upper bound on s
There is no (40, 52, 854871)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 2 220461 011749 668992 045000 833783 993065 > 552 [i]