Best Known (62, 62+13, s)-Nets in Base 7
(62, 62+13, 39217)-Net over F7 — Constructive and digital
Digital (62, 75, 39217)-net over F7, using
- 71 times duplication [i] based on digital (61, 74, 39217)-net over F7, using
- net defined by OOA [i] based on linear OOA(774, 39217, F7, 13, 13) (dual of [(39217, 13), 509747, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(774, 235303, F7, 13) (dual of [235303, 235229, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(774, 235304, F7, 13) (dual of [235304, 235230, 14]-code), using
- trace code [i] based on linear OA(4937, 117652, F49, 13) (dual of [117652, 117615, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(4937, 117649, F49, 13) (dual of [117649, 117612, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(490, 3, F49, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- trace code [i] based on linear OA(4937, 117652, F49, 13) (dual of [117652, 117615, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(774, 235304, F7, 13) (dual of [235304, 235230, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(774, 235303, F7, 13) (dual of [235303, 235229, 14]-code), using
- net defined by OOA [i] based on linear OOA(774, 39217, F7, 13, 13) (dual of [(39217, 13), 509747, 14]-NRT-code), using
(62, 62+13, 235306)-Net over F7 — Digital
Digital (62, 75, 235306)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(775, 235306, F7, 13) (dual of [235306, 235231, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(774, 235304, F7, 13) (dual of [235304, 235230, 14]-code), using
- trace code [i] based on linear OA(4937, 117652, F49, 13) (dual of [117652, 117615, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(4937, 117649, F49, 13) (dual of [117649, 117612, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(490, 3, F49, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- trace code [i] based on linear OA(4937, 117652, F49, 13) (dual of [117652, 117615, 14]-code), using
- linear OA(774, 235305, F7, 12) (dual of [235305, 235231, 13]-code), using Gilbert–Varšamov bound and bm = 774 > Vbs−1(k−1) = 1 112582 733665 983148 887713 491969 761991 730743 514477 866661 927361 [i]
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(774, 235304, F7, 13) (dual of [235304, 235230, 14]-code), using
- construction X with Varšamov bound [i] based on
(62, 62+13, large)-Net in Base 7 — Upper bound on s
There is no (62, 75, large)-net in base 7, because
- 11 times m-reduction [i] would yield (62, 64, large)-net in base 7, but