Best Known (77−13, 77, s)-Nets in Base 7
(77−13, 77, 39219)-Net over F7 — Constructive and digital
Digital (64, 77, 39219)-net over F7, using
- net defined by OOA [i] based on linear OOA(777, 39219, F7, 13, 13) (dual of [(39219, 13), 509770, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(777, 235315, F7, 13) (dual of [235315, 235238, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(776, 235314, F7, 13) (dual of [235314, 235238, 14]-code), using
- trace code [i] based on linear OA(4938, 117657, F49, 13) (dual of [117657, 117619, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(4937, 117650, F49, 13) (dual of [117650, 117613, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4931, 117650, F49, 11) (dual of [117650, 117619, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [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 C([0,6]) ⊂ C([0,5]) [i] based on
- trace code [i] based on linear OA(4938, 117657, F49, 13) (dual of [117657, 117619, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(776, 235314, F7, 13) (dual of [235314, 235238, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(777, 235315, F7, 13) (dual of [235315, 235238, 14]-code), using
(77−13, 77, 235316)-Net over F7 — Digital
Digital (64, 77, 235316)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(777, 235316, F7, 13) (dual of [235316, 235239, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(776, 235314, F7, 13) (dual of [235314, 235238, 14]-code), using
- trace code [i] based on linear OA(4938, 117657, F49, 13) (dual of [117657, 117619, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(4937, 117650, F49, 13) (dual of [117650, 117613, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4931, 117650, F49, 11) (dual of [117650, 117619, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [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 C([0,6]) ⊂ C([0,5]) [i] based on
- trace code [i] based on linear OA(4938, 117657, F49, 13) (dual of [117657, 117619, 14]-code), using
- linear OA(776, 235315, F7, 12) (dual of [235315, 235239, 13]-code), using Gilbert–Varšamov bound and bm = 776 > Vbs−1(k−1) = 1 113102 965470 602409 676983 885342 570968 825662 114260 117236 545713 [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(776, 235314, F7, 13) (dual of [235314, 235238, 14]-code), using
- construction X with Varšamov bound [i] based on
(77−13, 77, large)-Net in Base 7 — Upper bound on s
There is no (64, 77, large)-net in base 7, because
- 11 times m-reduction [i] would yield (64, 66, large)-net in base 7, but