Best Known (70−27, 70, s)-Nets in Base 7
(70−27, 70, 116)-Net over F7 — Constructive and digital
Digital (43, 70, 116)-net over F7, using
- trace code for nets [i] based on digital (8, 35, 58)-net over F49, using
- net from sequence [i] based on digital (8, 57)-sequence over F49, using
(70−27, 70, 348)-Net over F7 — Digital
Digital (43, 70, 348)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(770, 348, F7, 27) (dual of [348, 278, 28]-code), using
- construction XX applied to C1 = C([341,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([341,25]) [i] based on
- linear OA(767, 342, F7, 26) (dual of [342, 275, 27]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,24}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(767, 342, F7, 26) (dual of [342, 275, 27]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(770, 342, F7, 27) (dual of [342, 272, 28]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,25}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(764, 342, F7, 25) (dual of [342, 278, 26]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,24], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(70, 3, F7, 0) (dual of [3, 3, 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(70, 3, F7, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([341,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([341,25]) [i] based on
(70−27, 70, 28883)-Net in Base 7 — Upper bound on s
There is no (43, 70, 28884)-net in base 7, because
- 1 times m-reduction [i] would yield (43, 69, 28884)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 20503 463886 377953 230201 426651 622922 587835 070664 393374 426649 > 769 [i]