Best Known (83−27, 83, s)-Nets in Base 25
(83−27, 83, 1203)-Net over F25 — Constructive and digital
Digital (56, 83, 1203)-net over F25, using
- 253 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
(83−27, 83, 15651)-Net over F25 — Digital
Digital (56, 83, 15651)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2583, 15651, F25, 27) (dual of [15651, 15568, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(18) [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(2555, 15625, F25, 19) (dual of [15625, 15570, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(257, 26, F25, 7) (dual of [26, 19, 8]-code or 26-arc in PG(6,25)), using
- extended Reed–Solomon code RSe(19,25) [i]
- the expurgated narrow-sense BCH-code C(I) with length 26 | 252−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- algebraic-geometric code AG(F,9P) with degPÂ =Â 2 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using the rational function field F25(x) [i]
- algebraic-geometric code AG(F,6P) with degPÂ =Â 3 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26 (see above)
- construction X applied to Ce(26) ⊂ Ce(18) [i] based on
(83−27, 83, large)-Net in Base 25 — Upper bound on s
There is no (56, 83, large)-net in base 25, because
- 25 times m-reduction [i] would yield (56, 58, large)-net in base 25, but