Best Known (118, 118+27, s)-Nets in Base 4
(118, 118+27, 1261)-Net over F4 — Constructive and digital
Digital (118, 145, 1261)-net over F4, using
- 43 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(4142, 1261, F4, 27, 27) (dual of [(1261, 27), 33905, 28]-NRT-code), using
(118, 118+27, 9943)-Net over F4 — Digital
Digital (118, 145, 9943)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4145, 9943, F4, 27) (dual of [9943, 9798, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4145, 16404, F4, 27) (dual of [16404, 16259, 28]-code), using
- construction XX applied to Ce(26) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(4141, 16384, F4, 27) (dual of [16384, 16243, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4127, 16384, F4, 25) (dual of [16384, 16257, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(41, 17, F4, 1) (dual of [17, 16, 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
- linear OA(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- construction XX applied to Ce(26) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4145, 16404, F4, 27) (dual of [16404, 16259, 28]-code), using
(118, 118+27, large)-Net in Base 4 — Upper bound on s
There is no (118, 145, large)-net in base 4, because
- 25 times m-reduction [i] would yield (118, 120, large)-net in base 4, but