Best Known (73, 73+13, s)-Nets in Base 4
(73, 73+13, 43694)-Net over F4 — Constructive and digital
Digital (73, 86, 43694)-net over F4, using
- 41 times duplication [i] based on digital (72, 85, 43694)-net over F4, using
- net defined by OOA [i] based on linear OOA(485, 43694, F4, 13, 13) (dual of [(43694, 13), 567937, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(485, 262165, F4, 13) (dual of [262165, 262080, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(485, 262165, F4, 13) (dual of [262165, 262080, 14]-code), using
- net defined by OOA [i] based on linear OOA(485, 43694, F4, 13, 13) (dual of [(43694, 13), 567937, 14]-NRT-code), using
(73, 73+13, 131083)-Net over F4 — Digital
Digital (73, 86, 131083)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(486, 131083, F4, 2, 13) (dual of [(131083, 2), 262080, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(486, 262166, F4, 13) (dual of [262166, 262080, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(485, 262165, F4, 13) (dual of [262165, 262080, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(485, 262165, F4, 13) (dual of [262165, 262080, 14]-code), using
- OOA 2-folding [i] based on linear OA(486, 262166, F4, 13) (dual of [262166, 262080, 14]-code), using
(73, 73+13, large)-Net in Base 4 — Upper bound on s
There is no (73, 86, large)-net in base 4, because
- 11 times m-reduction [i] would yield (73, 75, large)-net in base 4, but