Best Known (172−24, 172, s)-Nets in Base 4
(172−24, 172, 21849)-Net over F4 — Constructive and digital
Digital (148, 172, 21849)-net over F4, using
- 1 times m-reduction [i] based on digital (148, 173, 21849)-net over F4, using
- net defined by OOA [i] based on linear OOA(4173, 21849, F4, 25, 25) (dual of [(21849, 25), 546052, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4173, 262189, F4, 25) (dual of [262189, 262016, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4173, 262191, F4, 25) (dual of [262191, 262018, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4127, 262145, F4, 19) (dual of [262145, 262018, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(410, 46, F4, 5) (dual of [46, 36, 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 C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4173, 262191, F4, 25) (dual of [262191, 262018, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4173, 262189, F4, 25) (dual of [262189, 262016, 26]-code), using
- net defined by OOA [i] based on linear OOA(4173, 21849, F4, 25, 25) (dual of [(21849, 25), 546052, 26]-NRT-code), using
(172−24, 172, 144320)-Net over F4 — Digital
Digital (148, 172, 144320)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4172, 144320, F4, 24) (dual of [144320, 144148, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4172, 262190, F4, 24) (dual of [262190, 262018, 25]-code), using
- 1 times truncation [i] based on linear OA(4173, 262191, F4, 25) (dual of [262191, 262018, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4127, 262145, F4, 19) (dual of [262145, 262018, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(410, 46, F4, 5) (dual of [46, 36, 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 C([0,12]) ⊂ C([0,9]) [i] based on
- 1 times truncation [i] based on linear OA(4173, 262191, F4, 25) (dual of [262191, 262018, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4172, 262190, F4, 24) (dual of [262190, 262018, 25]-code), using
(172−24, 172, large)-Net in Base 4 — Upper bound on s
There is no (148, 172, large)-net in base 4, because
- 22 times m-reduction [i] would yield (148, 150, large)-net in base 4, but