Best Known (58, 58+10, s)-Nets in Base 4
(58, 58+10, 52433)-Net over F4 — Constructive and digital
Digital (58, 68, 52433)-net over F4, using
- 41 times duplication [i] based on digital (57, 67, 52433)-net over F4, using
- net defined by OOA [i] based on linear OOA(467, 52433, F4, 10, 10) (dual of [(52433, 10), 524263, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(467, 262165, F4, 10) (dual of [262165, 262098, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(464, 262144, F4, 10) (dual of [262144, 262080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(446, 262144, F4, 7) (dual of [262144, 262098, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- OA 5-folding and stacking [i] based on linear OA(467, 262165, F4, 10) (dual of [262165, 262098, 11]-code), using
- net defined by OOA [i] based on linear OOA(467, 52433, F4, 10, 10) (dual of [(52433, 10), 524263, 11]-NRT-code), using
(58, 58+10, 138294)-Net over F4 — Digital
Digital (58, 68, 138294)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(468, 138294, F4, 10) (dual of [138294, 138226, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(468, 262166, F4, 10) (dual of [262166, 262098, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(467, 262165, F4, 10) (dual of [262165, 262098, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(464, 262144, F4, 10) (dual of [262144, 262080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(446, 262144, F4, 7) (dual of [262144, 262098, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(467, 262165, F4, 10) (dual of [262165, 262098, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(468, 262166, F4, 10) (dual of [262166, 262098, 11]-code), using
(58, 58+10, large)-Net in Base 4 — Upper bound on s
There is no (58, 68, large)-net in base 4, because
- 8 times m-reduction [i] would yield (58, 60, large)-net in base 4, but