Best Known (63−17, 63, s)-Nets in Base 7
(63−17, 63, 302)-Net over F7 — Constructive and digital
Digital (46, 63, 302)-net over F7, using
- 72 times duplication [i] based on digital (44, 61, 302)-net over F7, using
- net defined by OOA [i] based on linear OOA(761, 302, F7, 17, 17) (dual of [(302, 17), 5073, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(761, 2417, F7, 17) (dual of [2417, 2356, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(757, 2401, F7, 17) (dual of [2401, 2344, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(745, 2401, F7, 13) (dual of [2401, 2356, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(74, 16, F7, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,7)), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(761, 2417, F7, 17) (dual of [2417, 2356, 18]-code), using
- net defined by OOA [i] based on linear OOA(761, 302, F7, 17, 17) (dual of [(302, 17), 5073, 18]-NRT-code), using
(63−17, 63, 2593)-Net over F7 — Digital
Digital (46, 63, 2593)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(763, 2593, F7, 17) (dual of [2593, 2530, 18]-code), using
- 182 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 6 times 0, 1, 18 times 0, 1, 46 times 0, 1, 106 times 0) [i] based on linear OA(757, 2405, F7, 17) (dual of [2405, 2348, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(757, 2401, F7, 17) (dual of [2401, 2344, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(753, 2401, F7, 16) (dual of [2401, 2348, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 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(16) ⊂ Ce(15) [i] based on
- 182 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 6 times 0, 1, 18 times 0, 1, 46 times 0, 1, 106 times 0) [i] based on linear OA(757, 2405, F7, 17) (dual of [2405, 2348, 18]-code), using
(63−17, 63, 2223554)-Net in Base 7 — Upper bound on s
There is no (46, 63, 2223555)-net in base 7, because
- 1 times m-reduction [i] would yield (46, 62, 2223555)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 24893 160652 544065 419152 176141 880030 654701 152887 159377 > 762 [i]