Best Known (46, 55, s)-Nets in Base 7
(46, 55, 205900)-Net over F7 — Constructive and digital
Digital (46, 55, 205900)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 13)-net over F7, using
- 8 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (41, 50, 205887)-net over F7, using
- net defined by OOA [i] based on linear OOA(750, 205887, F7, 9, 9) (dual of [(205887, 9), 1852933, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(750, 823549, F7, 9) (dual of [823549, 823499, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(750, 823550, F7, 9) (dual of [823550, 823500, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(750, 823543, F7, 9) (dual of [823543, 823493, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 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
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(750, 823550, F7, 9) (dual of [823550, 823500, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(750, 823549, F7, 9) (dual of [823549, 823499, 10]-code), using
- net defined by OOA [i] based on linear OOA(750, 205887, F7, 9, 9) (dual of [(205887, 9), 1852933, 10]-NRT-code), using
- digital (1, 5, 13)-net over F7, using
(46, 55, 823570)-Net over F7 — Digital
Digital (46, 55, 823570)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(755, 823570, F7, 9) (dual of [823570, 823515, 10]-code), using
- construction XX applied to Ce(8) ⊂ Ce(4) ⊂ Ce(3) [i] based on
- linear OA(750, 823543, F7, 9) (dual of [823543, 823493, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(729, 823543, F7, 5) (dual of [823543, 823514, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(722, 823543, F7, 4) (dual of [823543, 823521, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(74, 26, F7, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,7)), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(8) ⊂ Ce(4) ⊂ Ce(3) [i] based on
(46, 55, large)-Net in Base 7 — Upper bound on s
There is no (46, 55, large)-net in base 7, because
- 7 times m-reduction [i] would yield (46, 48, large)-net in base 7, but