Best Known (16, 16+5, s)-Nets in Base 7
(16, 16+5, 9411)-Net over F7 — Constructive and digital
Digital (16, 21, 9411)-net over F7, using
- net defined by OOA [i] based on linear OOA(721, 9411, F7, 6, 5) (dual of [(9411, 6), 56445, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(721, 9412, F7, 2, 5) (dual of [(9412, 2), 18803, 6]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(71, 2353, F7, 2, 1) (dual of [(2353, 2), 4705, 2]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(71, s, F7, 2, 1) with arbitrarily large s, using
- appending 1 arbitrary column [i] based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- discarding factors / shortening the dual code based on linear OOA(71, s, F7, 2, 1) with arbitrarily large s, using
- linear OOA(71, 2353, F7, 2, 1) (dual of [(2353, 2), 4705, 2]-NRT-code) (see above)
- linear OOA(75, 2353, F7, 2, 2) (dual of [(2353, 2), 4701, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(75, 2801, F7, 2, 2) (dual of [(2801, 2), 5597, 3]-NRT-code), using
- appending kth column [i] based on linear OA(75, 2801, F7, 2) (dual of [2801, 2796, 3]-code), using
- Hamming code H(5,7) [i]
- appending kth column [i] based on linear OA(75, 2801, F7, 2) (dual of [2801, 2796, 3]-code), using
- discarding factors / shortening the dual code based on linear OOA(75, 2801, F7, 2, 2) (dual of [(2801, 2), 5597, 3]-NRT-code), using
- linear OOA(714, 2353, F7, 2, 5) (dual of [(2353, 2), 4692, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(714, 4706, F7, 5) (dual of [4706, 4692, 6]-code), using
- trace code [i] based on linear OA(497, 2353, F49, 5) (dual of [2353, 2346, 6]-code), using
- OOA 2-folding [i] based on linear OA(714, 4706, F7, 5) (dual of [4706, 4692, 6]-code), using
- linear OOA(71, 2353, F7, 2, 1) (dual of [(2353, 2), 4705, 2]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(721, 9412, F7, 2, 5) (dual of [(9412, 2), 18803, 6]-NRT-code), using
(16, 16+5, 16812)-Net over F7 — Digital
Digital (16, 21, 16812)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(721, 16812, F7, 5) (dual of [16812, 16791, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(721, 16807, F7, 5) (dual of [16807, 16786, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(716, 16807, F7, 4) (dual of [16807, 16791, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(70, 5, F7, 0) (dual of [5, 5, 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(4) ⊂ Ce(3) [i] based on
(16, 16+5, large)-Net in Base 7 — Upper bound on s
There is no (16, 21, large)-net in base 7, because
- 3 times m-reduction [i] would yield (16, 18, large)-net in base 7, but