Best Known (173−27, 173, s)-Nets in Base 4
(173−27, 173, 5045)-Net over F4 — Constructive and digital
Digital (146, 173, 5045)-net over F4, using
- 42 times duplication [i] based on digital (144, 171, 5045)-net over F4, using
- net defined by OOA [i] based on linear OOA(4171, 5045, F4, 27, 27) (dual of [(5045, 27), 136044, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4171, 65586, F4, 27) (dual of [65586, 65415, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- linear OA(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(410, 50, F4, 5) (dual of [50, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- OOA 13-folding and stacking with additional row [i] based on linear OA(4171, 65586, F4, 27) (dual of [65586, 65415, 28]-code), using
- net defined by OOA [i] based on linear OOA(4171, 5045, F4, 27, 27) (dual of [(5045, 27), 136044, 28]-NRT-code), using
(173−27, 173, 47043)-Net over F4 — Digital
Digital (146, 173, 47043)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4173, 47043, F4, 27) (dual of [47043, 46870, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4173, 65550, F4, 27) (dual of [65550, 65377, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([1,13]) [i] based on
- linear OA(4161, 65537, F4, 27) (dual of [65537, 65376, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(4160, 65537, F4, 14) (dual of [65537, 65377, 15]-code), using the narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [1,13], and minimum distance d ≥ |{−13,−11,−9,…,13}|+1 = 15 (BCH-bound) [i]
- linear OA(412, 13, F4, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,4)), using
- dual of repetition code with length 13 [i]
- construction X applied to C([0,13]) ⊂ C([1,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4173, 65550, F4, 27) (dual of [65550, 65377, 28]-code), using
(173−27, 173, large)-Net in Base 4 — Upper bound on s
There is no (146, 173, large)-net in base 4, because
- 25 times m-reduction [i] would yield (146, 148, large)-net in base 4, but