Best Known (75, 75+23, s)-Nets in Base 7
(75, 75+23, 1528)-Net over F7 — Constructive and digital
Digital (75, 98, 1528)-net over F7, using
- 72 times duplication [i] based on digital (73, 96, 1528)-net over F7, using
- net defined by OOA [i] based on linear OOA(796, 1528, F7, 23, 23) (dual of [(1528, 23), 35048, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(796, 16809, F7, 23) (dual of [16809, 16713, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(796, 16812, F7, 23) (dual of [16812, 16716, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(796, 16807, F7, 23) (dual of [16807, 16711, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(791, 16807, F7, 22) (dual of [16807, 16716, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(70, 5, F7, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(796, 16812, F7, 23) (dual of [16812, 16716, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(796, 16809, F7, 23) (dual of [16809, 16713, 24]-code), using
- net defined by OOA [i] based on linear OOA(796, 1528, F7, 23, 23) (dual of [(1528, 23), 35048, 24]-NRT-code), using
(75, 75+23, 11572)-Net over F7 — Digital
Digital (75, 98, 11572)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(798, 11572, F7, 23) (dual of [11572, 11474, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(798, 16815, F7, 23) (dual of [16815, 16717, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(796, 16807, F7, 23) (dual of [16807, 16711, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(786, 16807, F7, 20) (dual of [16807, 16721, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(72, 8, F7, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,7)), using
- extended Reed–Solomon code RSe(6,7) [i]
- Hamming code H(2,7) [i]
- algebraic-geometric code AG(F, Q+1P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using the rational function field F7(x) [i]
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(798, 16815, F7, 23) (dual of [16815, 16717, 24]-code), using
(75, 75+23, large)-Net in Base 7 — Upper bound on s
There is no (75, 98, large)-net in base 7, because
- 21 times m-reduction [i] would yield (75, 77, large)-net in base 7, but