Best Known (68, 68+14, s)-Nets in Base 7
(68, 68+14, 33616)-Net over F7 — Constructive and digital
Digital (68, 82, 33616)-net over F7, using
- net defined by OOA [i] based on linear OOA(782, 33616, F7, 14, 14) (dual of [(33616, 14), 470542, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(782, 235312, F7, 14) (dual of [235312, 235230, 15]-code), using
- trace code [i] based on linear OA(4941, 117656, F49, 14) (dual of [117656, 117615, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(4940, 117649, F49, 14) (dual of [117649, 117609, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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(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(13) ⊂ Ce(11) [i] based on
- trace code [i] based on linear OA(4941, 117656, F49, 14) (dual of [117656, 117615, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(782, 235312, F7, 14) (dual of [235312, 235230, 15]-code), using
(68, 68+14, 235312)-Net over F7 — Digital
Digital (68, 82, 235312)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(782, 235312, F7, 14) (dual of [235312, 235230, 15]-code), using
- trace code [i] based on linear OA(4941, 117656, F49, 14) (dual of [117656, 117615, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(4940, 117649, F49, 14) (dual of [117649, 117609, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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(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(13) ⊂ Ce(11) [i] based on
- trace code [i] based on linear OA(4941, 117656, F49, 14) (dual of [117656, 117615, 15]-code), using
(68, 68+14, large)-Net in Base 7 — Upper bound on s
There is no (68, 82, large)-net in base 7, because
- 12 times m-reduction [i] would yield (68, 70, large)-net in base 7, but