Best Known (142−27, 142, s)-Nets in Base 4
(142−27, 142, 1261)-Net over F4 — Constructive and digital
Digital (115, 142, 1261)-net over F4, using
- net defined by OOA [i] based on linear OOA(4142, 1261, F4, 27, 27) (dual of [(1261, 27), 33905, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4142, 16394, F4, 27) (dual of [16394, 16252, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4142, 16400, F4, 27) (dual of [16400, 16258, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(4141, 16385, F4, 27) (dual of [16385, 16244, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(4127, 16385, F4, 25) (dual of [16385, 16258, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(41, 15, F4, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4142, 16400, F4, 27) (dual of [16400, 16258, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4142, 16394, F4, 27) (dual of [16394, 16252, 28]-code), using
(142−27, 142, 8416)-Net over F4 — Digital
Digital (115, 142, 8416)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4142, 8416, F4, 27) (dual of [8416, 8274, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4142, 16400, F4, 27) (dual of [16400, 16258, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(4141, 16385, F4, 27) (dual of [16385, 16244, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(4127, 16385, F4, 25) (dual of [16385, 16258, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(41, 15, F4, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4142, 16400, F4, 27) (dual of [16400, 16258, 28]-code), using
(142−27, 142, 6402056)-Net in Base 4 — Upper bound on s
There is no (115, 142, 6402057)-net in base 4, because
- 1 times m-reduction [i] would yield (115, 141, 6402057)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 7 770681 029251 972561 072549 162904 068912 655927 502291 363928 488608 867379 015597 053042 375680 > 4141 [i]