Best Known (50, 50+16, s)-Nets in Base 25
(50, 50+16, 48831)-Net over F25 — Constructive and digital
Digital (50, 66, 48831)-net over F25, using
- 251 times duplication [i] based on digital (49, 65, 48831)-net over F25, using
- net defined by OOA [i] based on linear OOA(2565, 48831, F25, 16, 16) (dual of [(48831, 16), 781231, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(2565, 390648, F25, 16) (dual of [390648, 390583, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2565, 390649, F25, 16) (dual of [390649, 390584, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- 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(2541, 390625, F25, 11) (dual of [390625, 390584, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(254, 24, F25, 4) (dual of [24, 20, 5]-code or 24-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(15) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(2565, 390649, F25, 16) (dual of [390649, 390584, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(2565, 390648, F25, 16) (dual of [390648, 390583, 17]-code), using
- net defined by OOA [i] based on linear OOA(2565, 48831, F25, 16, 16) (dual of [(48831, 16), 781231, 17]-NRT-code), using
(50, 50+16, 390651)-Net over F25 — Digital
Digital (50, 66, 390651)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2566, 390651, F25, 16) (dual of [390651, 390585, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(9) [i] based on
- 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(2537, 390625, F25, 10) (dual of [390625, 390588, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(15) ⊂ Ce(9) [i] based on
(50, 50+16, large)-Net in Base 25 — Upper bound on s
There is no (50, 66, large)-net in base 25, because
- 14 times m-reduction [i] would yield (50, 52, large)-net in base 25, but