Best Known (57, 57+13, s)-Nets in Base 7
(57, 57+13, 19611)-Net over F7 — Constructive and digital
Digital (57, 70, 19611)-net over F7, using
- net defined by OOA [i] based on linear OOA(770, 19611, F7, 13, 13) (dual of [(19611, 13), 254873, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(770, 117667, F7, 13) (dual of [117667, 117597, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(770, 117670, F7, 13) (dual of [117670, 117600, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(767, 117649, F7, 13) (dual of [117649, 117582, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(749, 117649, F7, 10) (dual of [117649, 117600, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(770, 117670, F7, 13) (dual of [117670, 117600, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(770, 117667, F7, 13) (dual of [117667, 117597, 14]-code), using
(57, 57+13, 117670)-Net over F7 — Digital
Digital (57, 70, 117670)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(770, 117670, F7, 13) (dual of [117670, 117600, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(767, 117649, F7, 13) (dual of [117649, 117582, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(749, 117649, F7, 10) (dual of [117649, 117600, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(12) ⊂ Ce(9) [i] based on
(57, 57+13, large)-Net in Base 7 — Upper bound on s
There is no (57, 70, large)-net in base 7, because
- 11 times m-reduction [i] would yield (57, 59, large)-net in base 7, but