Best Known (100−33, 100, s)-Nets in Base 25
(100−33, 100, 978)-Net over F25 — Constructive and digital
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
(100−33, 100, 15058)-Net over F25 — Digital
Digital (67, 100, 15058)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25100, 15058, F25, 33) (dual of [15058, 14958, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(25100, 15641, F25, 33) (dual of [15641, 15541, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(2597, 15626, F25, 33) (dual of [15626, 15529, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(2585, 15626, F25, 29) (dual of [15626, 15541, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(253, 15, F25, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,25) or 15-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 C([0,16]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25100, 15641, F25, 33) (dual of [15641, 15541, 34]-code), using
(100−33, 100, large)-Net in Base 25 — Upper bound on s
There is no (67, 100, large)-net in base 25, because
- 31 times m-reduction [i] would yield (67, 69, large)-net in base 25, but