Best Known (64−18, 64, s)-Nets in Base 7
(64−18, 64, 268)-Net over F7 — Constructive and digital
Digital (46, 64, 268)-net over F7, using
- 71 times duplication [i] based on digital (45, 63, 268)-net over F7, using
- net defined by OOA [i] based on linear OOA(763, 268, F7, 18, 18) (dual of [(268, 18), 4761, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(763, 2412, F7, 18) (dual of [2412, 2349, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(15) ⊂ Ce(14) [i] based on
- 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(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(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(71, 10, F7, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(17) ⊂ Ce(15) ⊂ Ce(14) [i] based on
- OA 9-folding and stacking [i] based on linear OA(763, 2412, F7, 18) (dual of [2412, 2349, 19]-code), using
- net defined by OOA [i] based on linear OOA(763, 268, F7, 18, 18) (dual of [(268, 18), 4761, 19]-NRT-code), using
(64−18, 64, 2400)-Net over F7 — Digital
Digital (46, 64, 2400)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(764, 2400, F7, 18) (dual of [2400, 2336, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(764, 2416, F7, 18) (dual of [2416, 2352, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- 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(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(73, 15, F7, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(764, 2416, F7, 18) (dual of [2416, 2352, 19]-code), using
(64−18, 64, 706615)-Net in Base 7 — Upper bound on s
There is no (46, 64, 706616)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 1 219775 333226 788614 067886 617212 072039 976736 481132 745425 > 764 [i]