Best Known (31−7, 31, s)-Nets in Base 4
(31−7, 31, 1367)-Net over F4 — Constructive and digital
Digital (24, 31, 1367)-net over F4, using
- net defined by OOA [i] based on linear OOA(431, 1367, F4, 7, 7) (dual of [(1367, 7), 9538, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(431, 4102, F4, 7) (dual of [4102, 4071, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(431, 4096, F4, 7) (dual of [4096, 4065, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(425, 4096, F4, 6) (dual of [4096, 4071, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(431, 4102, F4, 7) (dual of [4102, 4071, 8]-code), using
(31−7, 31, 3554)-Net over F4 — Digital
Digital (24, 31, 3554)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(431, 3554, F4, 7) (dual of [3554, 3523, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(431, 4096, F4, 7) (dual of [4096, 4065, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(431, 4096, F4, 7) (dual of [4096, 4065, 8]-code), using
(31−7, 31, 635127)-Net in Base 4 — Upper bound on s
There is no (24, 31, 635128)-net in base 4, because
- 1 times m-reduction [i] would yield (24, 30, 635128)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1 152923 242245 846085 > 430 [i]