Best Known (41, 56, s)-Nets in Base 7
(41, 56, 351)-Net over F7 — Constructive and digital
Digital (41, 56, 351)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- digital (34, 49, 343)-net over F7, using
- net defined by OOA [i] based on linear OOA(749, 343, F7, 15, 15) (dual of [(343, 15), 5096, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [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]
- OOA 7-folding and stacking with additional row [i] based on linear OA(749, 2402, F7, 15) (dual of [2402, 2353, 16]-code), using
- net defined by OOA [i] based on linear OOA(749, 343, F7, 15, 15) (dual of [(343, 15), 5096, 16]-NRT-code), using
- digital (0, 7, 8)-net over F7, using
(41, 56, 2625)-Net over F7 — Digital
Digital (41, 56, 2625)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(756, 2625, F7, 15) (dual of [2625, 2569, 16]-code), using
- 213 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0, 1, 21 times 0, 1, 54 times 0, 1, 124 times 0) [i] based on linear OA(750, 2406, F7, 15) (dual of [2406, 2356, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [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(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(71, 5, F7, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- 213 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0, 1, 21 times 0, 1, 54 times 0, 1, 124 times 0) [i] based on linear OA(750, 2406, F7, 15) (dual of [2406, 2356, 16]-code), using
(41, 56, 2459363)-Net in Base 7 — Upper bound on s
There is no (41, 56, 2459364)-net in base 7, because
- 1 times m-reduction [i] would yield (41, 55, 2459364)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 30226 860919 662975 706582 169479 001802 466860 812641 > 755 [i]