Best Known (74−21, 74, s)-Nets in Base 7
(74−21, 74, 241)-Net over F7 — Constructive and digital
Digital (53, 74, 241)-net over F7, using
- net defined by OOA [i] based on linear OOA(774, 241, F7, 21, 21) (dual of [(241, 21), 4987, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(774, 2411, F7, 21) (dual of [2411, 2337, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(773, 2402, F7, 21) (dual of [2402, 2329, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(765, 2402, F7, 19) (dual of [2402, 2337, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(71, 9, F7, 1) (dual of [9, 8, 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 C([0,10]) ⊂ C([0,9]) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(774, 2411, F7, 21) (dual of [2411, 2337, 22]-code), using
(74−21, 74, 2322)-Net over F7 — Digital
Digital (53, 74, 2322)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(774, 2322, F7, 21) (dual of [2322, 2248, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(774, 2411, F7, 21) (dual of [2411, 2337, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(773, 2402, F7, 21) (dual of [2402, 2329, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(765, 2402, F7, 19) (dual of [2402, 2337, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(71, 9, F7, 1) (dual of [9, 8, 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 C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(774, 2411, F7, 21) (dual of [2411, 2337, 22]-code), using
(74−21, 74, 1114392)-Net in Base 7 — Upper bound on s
There is no (53, 74, 1114393)-net in base 7, because
- 1 times m-reduction [i] would yield (53, 73, 1114393)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 49 221773 482924 748727 034453 096395 868690 918212 714610 216787 946529 > 773 [i]