Best Known (67, 67+32, s)-Nets in Base 25
(67, 67+32, 978)-Net over F25 — Constructive and digital
Digital (67, 99, 978)-net over F25, using
- 1 times m-reduction [i] based on digital (67, 100, 978)-net over F25, using
- net defined by OOA [i] based on linear OOA(25100, 978, F25, 33, 33) (dual of [(978, 33), 32174, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(25100, 15649, F25, 33) (dual of [15649, 15549, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(25100, 15651, F25, 33) (dual of [15651, 15551, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(25) [i] based on
- linear OA(2594, 15625, F25, 33) (dual of [15625, 15531, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- 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(256, 26, F25, 6) (dual of [26, 20, 7]-code or 26-arc in PG(5,25)), using
- extended Reed–Solomon code RSe(20,25) [i]
- algebraic-geometric code AG(F, Q+8P) with degQ = 3 and 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+5P) with degQ = 4 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(32) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(25100, 15651, F25, 33) (dual of [15651, 15551, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(25100, 15649, F25, 33) (dual of [15649, 15549, 34]-code), using
- net defined by OOA [i] based on linear OOA(25100, 978, F25, 33, 33) (dual of [(978, 33), 32174, 34]-NRT-code), using
(67, 67+32, 15654)-Net over F25 — Digital
Digital (67, 99, 15654)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2599, 15654, F25, 32) (dual of [15654, 15555, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(23) [i] based on
- linear OA(2591, 15625, F25, 32) (dual of [15625, 15534, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2570, 15625, F25, 24) (dual of [15625, 15555, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(31) ⊂ Ce(23) [i] based on
(67, 67+32, large)-Net in Base 25 — Upper bound on s
There is no (67, 99, large)-net in base 25, because
- 30 times m-reduction [i] would yield (67, 69, large)-net in base 25, but