Best Known (37, 37+15, s)-Nets in Base 7
(37, 37+15, 344)-Net over F7 — Constructive and digital
Digital (37, 52, 344)-net over F7, using
- 71 times duplication [i] based on digital (36, 51, 344)-net over F7, using
- net defined by OOA [i] based on linear OOA(751, 344, F7, 15, 15) (dual of [(344, 15), 5109, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(751, 2409, F7, 15) (dual of [2409, 2358, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(749, 2401, F7, 15) (dual of [2401, 2352, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(741, 2401, F7, 12) (dual of [2401, 2360, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(14) ⊂ Ce(11) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(751, 2409, F7, 15) (dual of [2409, 2358, 16]-code), using
- net defined by OOA [i] based on linear OOA(751, 344, F7, 15, 15) (dual of [(344, 15), 5109, 16]-NRT-code), using
(37, 37+15, 1945)-Net over F7 — Digital
Digital (37, 52, 1945)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(752, 1945, F7, 15) (dual of [1945, 1893, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(752, 2410, F7, 15) (dual of [2410, 2358, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(749, 2402, F7, 15) (dual of [2402, 2353, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(741, 2402, F7, 11) (dual of [2402, 2361, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(73, 8, F7, 3) (dual of [8, 5, 4]-code or 8-arc in PG(2,7) or 8-cap in PG(2,7)), using
- extended Reed–Solomon code RSe(5,7) [i]
- oval in PG(2, 7) [i]
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(752, 2410, F7, 15) (dual of [2410, 2358, 16]-code), using
(37, 37+15, 808924)-Net in Base 7 — Upper bound on s
There is no (37, 52, 808925)-net in base 7, because
- 1 times m-reduction [i] would yield (37, 51, 808925)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 12 589302 766610 212972 357081 932535 740412 465111 > 751 [i]