Best Known (58, 58+14, s)-Nets in Base 4
(58, 58+14, 2341)-Net over F4 — Constructive and digital
Digital (58, 72, 2341)-net over F4, using
- 41 times duplication [i] based on digital (57, 71, 2341)-net over F4, using
- net defined by OOA [i] based on linear OOA(471, 2341, F4, 14, 14) (dual of [(2341, 14), 32703, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(471, 16387, F4, 14) (dual of [16387, 16316, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(471, 16391, F4, 14) (dual of [16391, 16320, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(471, 16384, F4, 14) (dual of [16384, 16313, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(471, 16391, F4, 14) (dual of [16391, 16320, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(471, 16387, F4, 14) (dual of [16387, 16316, 15]-code), using
- net defined by OOA [i] based on linear OOA(471, 2341, F4, 14, 14) (dual of [(2341, 14), 32703, 15]-NRT-code), using
(58, 58+14, 8196)-Net over F4 — Digital
Digital (58, 72, 8196)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(472, 8196, F4, 2, 14) (dual of [(8196, 2), 16320, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(472, 16392, F4, 14) (dual of [16392, 16320, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(471, 16391, F4, 14) (dual of [16391, 16320, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(471, 16384, F4, 14) (dual of [16384, 16313, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(471, 16391, F4, 14) (dual of [16391, 16320, 15]-code), using
- OOA 2-folding [i] based on linear OA(472, 16392, F4, 14) (dual of [16392, 16320, 15]-code), using
(58, 58+14, 1755549)-Net in Base 4 — Upper bound on s
There is no (58, 72, 1755550)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 22 300772 349800 622371 538147 425888 858348 883556 > 472 [i]