Best Known (111, 111+24, s)-Nets in Base 4
(111, 111+24, 1368)-Net over F4 — Constructive and digital
Digital (111, 135, 1368)-net over F4, using
- 1 times m-reduction [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
(111, 111+24, 14005)-Net over F4 — Digital
Digital (111, 135, 14005)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4135, 14005, F4, 24) (dual of [14005, 13870, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4135, 16421, F4, 24) (dual of [16421, 16286, 25]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4134, 16420, F4, 24) (dual of [16420, 16286, 25]-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(47, 35, F4, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4134, 16420, F4, 24) (dual of [16420, 16286, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4135, 16421, F4, 24) (dual of [16421, 16286, 25]-code), using
(111, 111+24, large)-Net in Base 4 — Upper bound on s
There is no (111, 135, large)-net in base 4, because
- 22 times m-reduction [i] would yield (111, 113, large)-net in base 4, but