Best Known (58, 58+13, s)-Nets in Base 7
(58, 58+13, 19612)-Net over F7 — Constructive and digital
Digital (58, 71, 19612)-net over F7, using
- net defined by OOA [i] based on linear OOA(771, 19612, F7, 13, 13) (dual of [(19612, 13), 254885, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(771, 117673, F7, 13) (dual of [117673, 117602, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(771, 117677, F7, 13) (dual of [117677, 117606, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [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(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(74, 28, F7, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,7)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(771, 117677, F7, 13) (dual of [117677, 117606, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(771, 117673, F7, 13) (dual of [117673, 117602, 14]-code), using
(58, 58+13, 117677)-Net over F7 — Digital
Digital (58, 71, 117677)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(771, 117677, F7, 13) (dual of [117677, 117606, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [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(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(74, 28, F7, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,7)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
(58, 58+13, large)-Net in Base 7 — Upper bound on s
There is no (58, 71, large)-net in base 7, because
- 11 times m-reduction [i] would yield (58, 60, large)-net in base 7, but