Best Known (57, 57+30, s)-Nets in Base 25
(57, 57+30, 1042)-Net over F25 — Constructive and digital
Digital (57, 87, 1042)-net over F25, using
- 251 times duplication [i] based on digital (56, 86, 1042)-net over F25, using
- net defined by OOA [i] based on linear OOA(2586, 1042, F25, 30, 30) (dual of [(1042, 30), 31174, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(2586, 15630, F25, 30) (dual of [15630, 15544, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2586, 15632, F25, 30) (dual of [15632, 15546, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(27) [i] based on
- linear OA(2585, 15625, F25, 30) (dual of [15625, 15540, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(2579, 15625, F25, 28) (dual of [15625, 15546, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(251, 7, F25, 1) (dual of [7, 6, 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 Ce(29) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(2586, 15632, F25, 30) (dual of [15632, 15546, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(2586, 15630, F25, 30) (dual of [15630, 15544, 31]-code), using
- net defined by OOA [i] based on linear OOA(2586, 1042, F25, 30, 30) (dual of [(1042, 30), 31174, 31]-NRT-code), using
(57, 57+30, 9243)-Net over F25 — Digital
Digital (57, 87, 9243)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2587, 9243, F25, 30) (dual of [9243, 9156, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2587, 15636, F25, 30) (dual of [15636, 15549, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(26) [i] based on
- linear OA(2585, 15625, F25, 30) (dual of [15625, 15540, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(2576, 15625, F25, 27) (dual of [15625, 15549, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(252, 11, F25, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,25)), using
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- Reed–Solomon code RS(23,25) [i]
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- construction X applied to Ce(29) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(2587, 15636, F25, 30) (dual of [15636, 15549, 31]-code), using
(57, 57+30, large)-Net in Base 25 — Upper bound on s
There is no (57, 87, large)-net in base 25, because
- 28 times m-reduction [i] would yield (57, 59, large)-net in base 25, but