Best Known (79, 79+27, s)-Nets in Base 25
(79, 79+27, 30049)-Net over F25 — Constructive and digital
Digital (79, 106, 30049)-net over F25, using
- 252 times duplication [i] based on digital (77, 104, 30049)-net over F25, using
- net defined by OOA [i] based on linear OOA(25104, 30049, F25, 27, 27) (dual of [(30049, 27), 811219, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(25104, 390638, F25, 27) (dual of [390638, 390534, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(25104, 390640, F25, 27) (dual of [390640, 390536, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(25101, 390625, F25, 27) (dual of [390625, 390524, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2589, 390625, F25, 23) (dual of [390625, 390536, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(253, 15, F25, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,25) or 15-cap in PG(2,25)), using
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- Reed–Solomon code RS(22,25) [i]
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(25104, 390640, F25, 27) (dual of [390640, 390536, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(25104, 390638, F25, 27) (dual of [390638, 390534, 28]-code), using
- net defined by OOA [i] based on linear OOA(25104, 30049, F25, 27, 27) (dual of [(30049, 27), 811219, 28]-NRT-code), using
(79, 79+27, 315316)-Net over F25 — Digital
Digital (79, 106, 315316)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25106, 315316, F25, 27) (dual of [315316, 315210, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(25106, 390635, F25, 27) (dual of [390635, 390529, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(25105, 390626, F25, 27) (dual of [390626, 390521, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(2597, 390626, F25, 25) (dual of [390626, 390529, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(251, 9, F25, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25106, 390635, F25, 27) (dual of [390635, 390529, 28]-code), using
(79, 79+27, large)-Net in Base 25 — Upper bound on s
There is no (79, 106, large)-net in base 25, because
- 25 times m-reduction [i] would yield (79, 81, large)-net in base 25, but