Best Known (36−7, 36, s)-Nets in Base 4
(36−7, 36, 5463)-Net over F4 — Constructive and digital
Digital (29, 36, 5463)-net over F4, using
- net defined by OOA [i] based on linear OOA(436, 5463, F4, 7, 7) (dual of [(5463, 7), 38205, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(436, 16390, F4, 7) (dual of [16390, 16354, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(436, 16391, F4, 7) (dual of [16391, 16355, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(436, 16384, F4, 7) (dual of [16384, 16348, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(429, 16384, F4, 6) (dual of [16384, 16355, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 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
- discarding factors / shortening the dual code based on linear OA(436, 16391, F4, 7) (dual of [16391, 16355, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(436, 16390, F4, 7) (dual of [16390, 16354, 8]-code), using
(36−7, 36, 14225)-Net over F4 — Digital
Digital (29, 36, 14225)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(436, 14225, F4, 7) (dual of [14225, 14189, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(436, 16384, F4, 7) (dual of [16384, 16348, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(436, 16384, F4, 7) (dual of [16384, 16348, 8]-code), using
(36−7, 36, 6401704)-Net in Base 4 — Upper bound on s
There is no (29, 36, 6401705)-net in base 4, because
- 1 times m-reduction [i] would yield (29, 35, 6401705)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1180 592154 498792 292696 > 435 [i]