Best Known (52, 52+12, s)-Nets in Base 7
(52, 52+12, 19611)-Net over F7 — Constructive and digital
Digital (52, 64, 19611)-net over F7, using
- net defined by OOA [i] based on linear OOA(764, 19611, F7, 12, 12) (dual of [(19611, 12), 235268, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(764, 117666, F7, 12) (dual of [117666, 117602, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(764, 117670, F7, 12) (dual of [117670, 117606, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(761, 117649, F7, 12) (dual of [117649, 117588, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(743, 117649, F7, 9) (dual of [117649, 117606, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(73, 21, F7, 2) (dual of [21, 18, 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(11) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(764, 117670, F7, 12) (dual of [117670, 117606, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(764, 117666, F7, 12) (dual of [117666, 117602, 13]-code), using
(52, 52+12, 117670)-Net over F7 — Digital
Digital (52, 64, 117670)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(764, 117670, F7, 12) (dual of [117670, 117606, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(761, 117649, F7, 12) (dual of [117649, 117588, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(743, 117649, F7, 9) (dual of [117649, 117606, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(73, 21, F7, 2) (dual of [21, 18, 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(11) ⊂ Ce(8) [i] based on
(52, 52+12, large)-Net in Base 7 — Upper bound on s
There is no (52, 64, large)-net in base 7, because
- 10 times m-reduction [i] would yield (52, 54, large)-net in base 7, but