Best Known (90−22, 90, s)-Nets in Base 25
(90−22, 90, 35513)-Net over F25 — Constructive and digital
Digital (68, 90, 35513)-net over F25, using
- 252 times duplication [i] based on digital (66, 88, 35513)-net over F25, using
- net defined by OOA [i] based on linear OOA(2588, 35513, F25, 22, 22) (dual of [(35513, 22), 781198, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(2588, 390643, F25, 22) (dual of [390643, 390555, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2588, 390644, F25, 22) (dual of [390644, 390556, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- linear OA(2585, 390625, F25, 22) (dual of [390625, 390540, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2569, 390625, F25, 18) (dual of [390625, 390556, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(253, 19, F25, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,25) or 19-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(21) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(2588, 390644, F25, 22) (dual of [390644, 390556, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(2588, 390643, F25, 22) (dual of [390643, 390555, 23]-code), using
- net defined by OOA [i] based on linear OOA(2588, 35513, F25, 22, 22) (dual of [(35513, 22), 781198, 23]-NRT-code), using
(90−22, 90, 390651)-Net over F25 — Digital
Digital (68, 90, 390651)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2590, 390651, F25, 22) (dual of [390651, 390561, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(2585, 390625, F25, 22) (dual of [390625, 390540, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2561, 390625, F25, 16) (dual of [390625, 390564, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(255, 26, F25, 5) (dual of [26, 21, 6]-code or 26-arc in PG(4,25)), using
- extended Reed–Solomon code RSe(21,25) [i]
- the expurgated narrow-sense BCH-code C(I) with length 26 | 252−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- algebraic-geometric code AG(F,10P) 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, Q+6P) with degQ = 2 and 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(21) ⊂ Ce(15) [i] based on
(90−22, 90, large)-Net in Base 25 — Upper bound on s
There is no (68, 90, large)-net in base 25, because
- 20 times m-reduction [i] would yield (68, 70, large)-net in base 25, but