Best Known (74, 74+13, s)-Nets in Base 5
(74, 74+13, 65111)-Net over F5 — Constructive and digital
Digital (74, 87, 65111)-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 (68, 81, 65105)-net over F5, using
- net defined by OOA [i] based on linear OOA(581, 65105, F5, 13, 13) (dual of [(65105, 13), 846284, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(581, 390631, F5, 13) (dual of [390631, 390550, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(581, 390633, F5, 13) (dual of [390633, 390552, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(581, 390625, F5, 13) (dual of [390625, 390544, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(573, 390625, F5, 12) (dual of [390625, 390552, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 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(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(581, 390633, F5, 13) (dual of [390633, 390552, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(581, 390631, F5, 13) (dual of [390631, 390550, 14]-code), using
- net defined by OOA [i] based on linear OOA(581, 65105, F5, 13, 13) (dual of [(65105, 13), 846284, 14]-NRT-code), using
- digital (0, 6, 6)-net over F5, using
(74, 74+13, 357783)-Net over F5 — Digital
Digital (74, 87, 357783)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(587, 357783, F5, 13) (dual of [357783, 357696, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(587, 390633, F5, 13) (dual of [390633, 390546, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([1,6]) [i] based on
- linear OA(581, 390626, F5, 13) (dual of [390626, 390545, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 516−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(580, 390626, F5, 6) (dual of [390626, 390546, 7]-code), using the narrow-sense BCH-code C(I) with length 390626 | 516−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(56, 7, F5, 6) (dual of [7, 1, 7]-code or 7-arc in PG(5,5)), using
- dual of repetition code with length 7 [i]
- construction X applied to C([0,6]) ⊂ C([1,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(587, 390633, F5, 13) (dual of [390633, 390546, 14]-code), using
(74, 74+13, large)-Net in Base 5 — Upper bound on s
There is no (74, 87, large)-net in base 5, because
- 11 times m-reduction [i] would yield (74, 76, large)-net in base 5, but