Best Known (48, 48+19, s)-Nets in Base 7
(48, 48+19, 267)-Net over F7 — Constructive and digital
Digital (48, 67, 267)-net over F7, using
- 72 times duplication [i] based on digital (46, 65, 267)-net over F7, using
- net defined by OOA [i] based on linear OOA(765, 267, F7, 19, 19) (dual of [(267, 19), 5008, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(765, 2404, F7, 19) (dual of [2404, 2339, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(765, 2405, F7, 19) (dual of [2405, 2340, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- 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(761, 2401, F7, 18) (dual of [2401, 2340, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(765, 2405, F7, 19) (dual of [2405, 2340, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(765, 2404, F7, 19) (dual of [2404, 2339, 20]-code), using
- net defined by OOA [i] based on linear OOA(765, 267, F7, 19, 19) (dual of [(267, 19), 5008, 20]-NRT-code), using
(48, 48+19, 2274)-Net over F7 — Digital
Digital (48, 67, 2274)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(767, 2274, F7, 19) (dual of [2274, 2207, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(767, 2412, F7, 19) (dual of [2412, 2345, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(766, 2411, F7, 19) (dual of [2411, 2345, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- 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(757, 2402, F7, 17) (dual of [2402, 2345, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,9]) ⊂ C([0,8]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(766, 2411, F7, 19) (dual of [2411, 2345, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(767, 2412, F7, 19) (dual of [2412, 2345, 20]-code), using
(48, 48+19, 1088889)-Net in Base 7 — Upper bound on s
There is no (48, 67, 1088890)-net in base 7, because
- 1 times m-reduction [i] would yield (48, 66, 1088890)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 59 768732 758564 953178 590601 944942 505445 213548 283463 470029 > 766 [i]