Best Known (39, 39+16, s)-Nets in Base 7
(39, 39+16, 301)-Net over F7 — Constructive and digital
Digital (39, 55, 301)-net over F7, using
- net defined by OOA [i] based on linear OOA(755, 301, F7, 16, 16) (dual of [(301, 16), 4761, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(755, 2408, F7, 16) (dual of [2408, 2353, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(755, 2409, F7, 16) (dual of [2409, 2354, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(753, 2401, F7, 16) (dual of [2401, 2348, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(745, 2401, F7, 13) (dual of [2401, 2356, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(755, 2409, F7, 16) (dual of [2409, 2354, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(755, 2408, F7, 16) (dual of [2408, 2353, 17]-code), using
(39, 39+16, 1824)-Net over F7 — Digital
Digital (39, 55, 1824)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(755, 1824, F7, 16) (dual of [1824, 1769, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(755, 2409, F7, 16) (dual of [2409, 2354, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(753, 2401, F7, 16) (dual of [2401, 2348, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(745, 2401, F7, 13) (dual of [2401, 2356, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(755, 2409, F7, 16) (dual of [2409, 2354, 17]-code), using
(39, 39+16, 405119)-Net in Base 7 — Upper bound on s
There is no (39, 55, 405120)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 30227 372811 180800 113369 765604 984470 124035 559425 > 755 [i]