Best Known (44, 44+16, s)-Nets in Base 7
(44, 44+16, 303)-Net over F7 — Constructive and digital
Digital (44, 60, 303)-net over F7, using
- net defined by OOA [i] based on linear OOA(760, 303, F7, 16, 16) (dual of [(303, 16), 4788, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(760, 2424, F7, 16) (dual of [2424, 2364, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(760, 2428, F7, 16) (dual of [2428, 2368, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(9) [i] based on
- 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(733, 2401, F7, 10) (dual of [2401, 2368, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(77, 27, F7, 5) (dual of [27, 20, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to Ce(15) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(760, 2428, F7, 16) (dual of [2428, 2368, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(760, 2424, F7, 16) (dual of [2424, 2364, 17]-code), using
(44, 44+16, 2700)-Net over F7 — Digital
Digital (44, 60, 2700)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(760, 2700, F7, 16) (dual of [2700, 2640, 17]-code), using
- 288 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 0, 1, 11 times 0, 1, 31 times 0, 1, 76 times 0, 1, 161 times 0) [i] based on linear OA(753, 2405, F7, 16) (dual of [2405, 2352, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- 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(749, 2401, F7, 15) (dual of [2401, 2352, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(15) ⊂ Ce(14) [i] based on
- 288 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 0, 1, 11 times 0, 1, 31 times 0, 1, 76 times 0, 1, 161 times 0) [i] based on linear OA(753, 2405, F7, 16) (dual of [2405, 2352, 17]-code), using
(44, 44+16, 1367012)-Net in Base 7 — Upper bound on s
There is no (44, 60, 1367013)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 508 022469 087463 730657 120556 758992 594951 255796 779777 > 760 [i]