Best Known (46, 62, s)-Nets in Base 7
(46, 62, 600)-Net over F7 — Constructive and digital
Digital (46, 62, 600)-net over F7, using
- net defined by OOA [i] based on linear OOA(762, 600, F7, 16, 16) (dual of [(600, 16), 9538, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(762, 4800, F7, 16) (dual of [4800, 4738, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(762, 4802, F7, 16) (dual of [4802, 4740, 17]-code), using
- trace code [i] based on linear OA(4931, 2401, F49, 16) (dual of [2401, 2370, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- trace code [i] based on linear OA(4931, 2401, F49, 16) (dual of [2401, 2370, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(762, 4802, F7, 16) (dual of [4802, 4740, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(762, 4800, F7, 16) (dual of [4800, 4738, 17]-code), using
(46, 62, 4806)-Net over F7 — Digital
Digital (46, 62, 4806)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(762, 4806, F7, 16) (dual of [4806, 4744, 17]-code), using
- trace code [i] based on linear OA(4931, 2403, F49, 16) (dual of [2403, 2372, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- linear OA(4931, 2401, F49, 16) (dual of [2401, 2370, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(4929, 2401, F49, 15) (dual of [2401, 2372, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(490, 2, F49, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- trace code [i] based on linear OA(4931, 2403, F49, 16) (dual of [2403, 2372, 17]-code), using
(46, 62, 2223554)-Net in Base 7 — Upper bound on s
There is no (46, 62, 2223555)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 24893 160652 544065 419152 176141 880030 654701 152887 159377 > 762 [i]