Best Known (82−27, 82, s)-Nets in Base 25
(82−27, 82, 1203)-Net over F25 — Constructive and digital
Digital (55, 82, 1203)-net over F25, using
- 252 times duplication [i] based on digital (53, 80, 1203)-net over F25, using
- net defined by OOA [i] based on linear OOA(2580, 1203, F25, 27, 27) (dual of [(1203, 27), 32401, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2580, 15640, F25, 27) (dual of [15640, 15560, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2580, 15641, F25, 27) (dual of [15641, 15561, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- 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(2564, 15625, F25, 22) (dual of [15625, 15561, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(254, 16, F25, 4) (dual of [16, 12, 5]-code or 16-arc in PG(3,25)), using
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- Reed–Solomon code RS(21,25) [i]
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(2580, 15641, F25, 27) (dual of [15641, 15561, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2580, 15640, F25, 27) (dual of [15640, 15560, 28]-code), using
- net defined by OOA [i] based on linear OOA(2580, 1203, F25, 27, 27) (dual of [(1203, 27), 32401, 28]-NRT-code), using
(82−27, 82, 14334)-Net over F25 — Digital
Digital (55, 82, 14334)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2582, 14334, F25, 27) (dual of [14334, 14252, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2582, 15641, F25, 27) (dual of [15641, 15559, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- linear OA(2579, 15626, F25, 27) (dual of [15626, 15547, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(2567, 15626, F25, 23) (dual of [15626, 15559, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [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 C([0,13]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2582, 15641, F25, 27) (dual of [15641, 15559, 28]-code), using
(82−27, 82, large)-Net in Base 25 — Upper bound on s
There is no (55, 82, large)-net in base 25, because
- 25 times m-reduction [i] would yield (55, 57, large)-net in base 25, but