Best Known (112, 112+25, s)-Nets in Base 4
(112, 112+25, 1368)-Net over F4 — Constructive and digital
Digital (112, 137, 1368)-net over F4, using
- 41 times duplication [i] based on digital (111, 136, 1368)-net over F4, using
- net defined by OOA [i] based on linear OOA(4136, 1368, F4, 25, 25) (dual of [(1368, 25), 34064, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4136, 16417, F4, 25) (dual of [16417, 16281, 26]-code), using
- construction XX applied to Ce(24) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- 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(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(499, 16384, F4, 19) (dual of [16384, 16285, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- construction XX applied to Ce(24) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(4136, 16417, F4, 25) (dual of [16417, 16281, 26]-code), using
- net defined by OOA [i] based on linear OOA(4136, 1368, F4, 25, 25) (dual of [(1368, 25), 34064, 26]-NRT-code), using
(112, 112+25, 11393)-Net over F4 — Digital
Digital (112, 137, 11393)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4137, 11393, F4, 25) (dual of [11393, 11256, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4137, 16423, F4, 25) (dual of [16423, 16286, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- 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(499, 16385, F4, 19) (dual of [16385, 16286, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(410, 38, F4, 5) (dual of [38, 28, 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(4137, 16423, F4, 25) (dual of [16423, 16286, 26]-code), using
(112, 112+25, large)-Net in Base 4 — Upper bound on s
There is no (112, 137, large)-net in base 4, because
- 23 times m-reduction [i] would yield (112, 114, large)-net in base 4, but