Best Known (90−29, 90, s)-Nets in Base 25
(90−29, 90, 1118)-Net over F25 — Constructive and digital
Digital (61, 90, 1118)-net over F25, using
- net defined by OOA [i] based on linear OOA(2590, 1118, F25, 29, 29) (dual of [(1118, 29), 32332, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2590, 15653, F25, 29) (dual of [15653, 15563, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2590, 15654, F25, 29) (dual of [15654, 15564, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- linear OA(2582, 15625, F25, 29) (dual of [15625, 15543, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2561, 15625, F25, 21) (dual of [15625, 15564, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(258, 29, F25, 7) (dual of [29, 21, 8]-code), using
- construction X applied to AG(F,9P) ⊂ AG(F,10P) [i] based on
- linear OA(257, 26, F25, 7) (dual of [26, 19, 8]-code or 26-arc in PG(6,25)), using 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]
- linear OA(255, 26, F25, 5) (dual of [26, 21, 6]-code or 26-arc in PG(4,25)), using 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 (see above)
- linear OA(251, 3, F25, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to AG(F,9P) ⊂ AG(F,10P) [i] based on
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2590, 15654, F25, 29) (dual of [15654, 15564, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2590, 15653, F25, 29) (dual of [15653, 15563, 30]-code), using
(90−29, 90, 15654)-Net over F25 — Digital
Digital (61, 90, 15654)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2590, 15654, F25, 29) (dual of [15654, 15564, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- linear OA(2582, 15625, F25, 29) (dual of [15625, 15543, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2561, 15625, F25, 21) (dual of [15625, 15564, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(258, 29, F25, 7) (dual of [29, 21, 8]-code), using
- construction X applied to AG(F,9P) ⊂ AG(F,10P) [i] based on
- linear OA(257, 26, F25, 7) (dual of [26, 19, 8]-code or 26-arc in PG(6,25)), using 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]
- linear OA(255, 26, F25, 5) (dual of [26, 21, 6]-code or 26-arc in PG(4,25)), using 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 (see above)
- linear OA(251, 3, F25, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to AG(F,9P) ⊂ AG(F,10P) [i] based on
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
(90−29, 90, large)-Net in Base 25 — Upper bound on s
There is no (61, 90, large)-net in base 25, because
- 27 times m-reduction [i] would yield (61, 63, large)-net in base 25, but