Best Known (53, 53+22, s)-Nets in Base 7
(53, 53+22, 219)-Net over F7 — Constructive and digital
Digital (53, 75, 219)-net over F7, using
- net defined by OOA [i] based on linear OOA(775, 219, F7, 22, 22) (dual of [(219, 22), 4743, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(775, 2409, F7, 22) (dual of [2409, 2334, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(773, 2401, F7, 22) (dual of [2401, 2328, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(765, 2401, F7, 19) (dual of [2401, 2336, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- OA 11-folding and stacking [i] based on linear OA(775, 2409, F7, 22) (dual of [2409, 2334, 23]-code), using
(53, 53+22, 1842)-Net over F7 — Digital
Digital (53, 75, 1842)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(775, 1842, F7, 22) (dual of [1842, 1767, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(775, 2409, F7, 22) (dual of [2409, 2334, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(773, 2401, F7, 22) (dual of [2401, 2328, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(765, 2401, F7, 19) (dual of [2401, 2336, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(775, 2409, F7, 22) (dual of [2409, 2334, 23]-code), using
(53, 53+22, 473030)-Net in Base 7 — Upper bound on s
There is no (53, 75, 473031)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 2411 918787 792427 117882 024662 711862 716504 062831 290982 152174 278271 > 775 [i]