Best Known (54, 54+26, s)-Nets in Base 25
(54, 54+26, 1203)-Net over F25 — Constructive and digital
Digital (54, 80, 1203)-net over F25, using
- t-expansion [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
(54, 54+26, 15651)-Net over F25 — Digital
Digital (54, 80, 15651)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2580, 15651, F25, 26) (dual of [15651, 15571, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(17) [i] based on
- linear OA(2573, 15625, F25, 26) (dual of [15625, 15552, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2552, 15625, F25, 18) (dual of [15625, 15573, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(25) ⊂ Ce(17) [i] based on
(54, 54+26, large)-Net in Base 25 — Upper bound on s
There is no (54, 80, large)-net in base 25, because
- 24 times m-reduction [i] would yield (54, 56, large)-net in base 25, but