Best Known (59, 59+28, s)-Nets in Base 25
(59, 59+28, 1118)-Net over F25 — Constructive and digital
Digital (59, 87, 1118)-net over F25, using
- net defined by OOA [i] based on linear OOA(2587, 1118, F25, 28, 28) (dual of [(1118, 28), 31217, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(2587, 15652, F25, 28) (dual of [15652, 15565, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2587, 15654, F25, 28) (dual of [15654, 15567, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(19) [i] based on
- linear OA(2579, 15625, F25, 28) (dual of [15625, 15546, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2558, 15625, F25, 20) (dual of [15625, 15567, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(27) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(2587, 15654, F25, 28) (dual of [15654, 15567, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(2587, 15652, F25, 28) (dual of [15652, 15565, 29]-code), using
(59, 59+28, 15654)-Net over F25 — Digital
Digital (59, 87, 15654)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2587, 15654, F25, 28) (dual of [15654, 15567, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(19) [i] based on
- linear OA(2579, 15625, F25, 28) (dual of [15625, 15546, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2558, 15625, F25, 20) (dual of [15625, 15567, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(27) ⊂ Ce(19) [i] based on
(59, 59+28, large)-Net in Base 25 — Upper bound on s
There is no (59, 87, large)-net in base 25, because
- 26 times m-reduction [i] would yield (59, 61, large)-net in base 25, but