Best Known (58, 58+12, s)-Nets in Base 7
(58, 58+12, 39218)-Net over F7 — Constructive and digital
Digital (58, 70, 39218)-net over F7, using
- net defined by OOA [i] based on linear OOA(770, 39218, F7, 12, 12) (dual of [(39218, 12), 470546, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(770, 235308, F7, 12) (dual of [235308, 235238, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(770, 235312, F7, 12) (dual of [235312, 235242, 13]-code), using
- trace code [i] based on linear OA(4935, 117656, F49, 12) (dual of [117656, 117621, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(4934, 117649, F49, 12) (dual of [117649, 117615, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(4928, 117649, F49, 10) (dual of [117649, 117621, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(491, 7, F49, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(4935, 117656, F49, 12) (dual of [117656, 117621, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(770, 235312, F7, 12) (dual of [235312, 235242, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(770, 235308, F7, 12) (dual of [235308, 235238, 13]-code), using
(58, 58+12, 235312)-Net over F7 — Digital
Digital (58, 70, 235312)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(770, 235312, F7, 12) (dual of [235312, 235242, 13]-code), using
- trace code [i] based on linear OA(4935, 117656, F49, 12) (dual of [117656, 117621, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(4934, 117649, F49, 12) (dual of [117649, 117615, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(4928, 117649, F49, 10) (dual of [117649, 117621, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(491, 7, F49, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(4935, 117656, F49, 12) (dual of [117656, 117621, 13]-code), using
(58, 58+12, large)-Net in Base 7 — Upper bound on s
There is no (58, 70, large)-net in base 7, because
- 10 times m-reduction [i] would yield (58, 60, large)-net in base 7, but