Best Known (29−6, 29, s)-Nets in Base 4
(29−6, 29, 5463)-Net over F4 — Constructive and digital
Digital (23, 29, 5463)-net over F4, using
- net defined by OOA [i] based on linear OOA(429, 5463, F4, 6, 6) (dual of [(5463, 6), 32749, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(429, 16389, F4, 6) (dual of [16389, 16360, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(429, 16391, F4, 6) (dual of [16391, 16362, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- 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(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(429, 16391, F4, 6) (dual of [16391, 16362, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(429, 16389, F4, 6) (dual of [16389, 16360, 7]-code), using
(29−6, 29, 12086)-Net over F4 — Digital
Digital (23, 29, 12086)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(429, 12086, F4, 6) (dual of [12086, 12057, 7]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(429, 16384, F4, 6) (dual of [16384, 16355, 7]-code), using
(29−6, 29, 400104)-Net in Base 4 — Upper bound on s
There is no (23, 29, 400105)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 288231 181821 110296 > 429 [i]