Best Known (42, 42+7, s)-Nets in Base 5
(42, 42+7, 651050)-Net over F5 — Constructive and digital
Digital (42, 49, 651050)-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 (39, 46, 651044)-net over F5, using
- net defined by OOA [i] based on linear OOA(546, 651044, F5, 7, 7) (dual of [(651044, 7), 4557262, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(546, 1953133, F5, 7) (dual of [1953133, 1953087, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(546, 1953134, F5, 7) (dual of [1953134, 1953088, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(546, 1953125, F5, 7) (dual of [1953125, 1953079, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(537, 1953125, F5, 6) (dual of [1953125, 1953088, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(50, 9, F5, 0) (dual of [9, 9, 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
- discarding factors / shortening the dual code based on linear OA(546, 1953134, F5, 7) (dual of [1953134, 1953088, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(546, 1953133, F5, 7) (dual of [1953133, 1953087, 8]-code), using
- net defined by OOA [i] based on linear OOA(546, 651044, F5, 7, 7) (dual of [(651044, 7), 4557262, 8]-NRT-code), using
- digital (0, 3, 6)-net over F5, using
(42, 42+7, 1953146)-Net over F5 — Digital
Digital (42, 49, 1953146)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(549, 1953146, F5, 7) (dual of [1953146, 1953097, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(546, 1953125, F5, 7) (dual of [1953125, 1953079, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(528, 1953125, F5, 4) (dual of [1953125, 1953097, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(53, 21, F5, 2) (dual of [21, 18, 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
(42, 42+7, large)-Net in Base 5 — Upper bound on s
There is no (42, 49, large)-net in base 5, because
- 5 times m-reduction [i] would yield (42, 44, large)-net in base 5, but