Best Known (69, 69+33, s)-Nets in Base 25
(69, 69+33, 978)-Net over F25 — Constructive and digital
Digital (69, 102, 978)-net over F25, using
- 252 times duplication [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
(69, 69+33, 15655)-Net over F25 — Digital
Digital (69, 102, 15655)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25102, 15655, F25, 33) (dual of [15655, 15553, 34]-code), using
- construction X with Varšamov bound [i] 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
- linear OA(25100, 15653, F25, 32) (dual of [15653, 15553, 33]-code), using Gilbert–Varšamov bound and bm = 25100 > Vbs−1(k−1) = 7 768138 372170 105658 498765 128949 951032 613529 708380 511366 428590 129247 516026 399492 678755 915986 561544 413425 927146 142459 090943 144738 016431 590625 [i]
- linear OA(250, 2, F25, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(25100, 15651, F25, 33) (dual of [15651, 15551, 34]-code), using
- construction X with Varšamov bound [i] based on
(69, 69+33, large)-Net in Base 25 — Upper bound on s
There is no (69, 102, large)-net in base 25, because
- 31 times m-reduction [i] would yield (69, 71, large)-net in base 25, but