Best Known (104, 104+25, s)-Nets in Base 5
(104, 104+25, 1304)-Net over F5 — Constructive and digital
Digital (104, 129, 1304)-net over F5, using
- 54 times duplication [i] based on digital (100, 125, 1304)-net over F5, using
- net defined by OOA [i] based on linear OOA(5125, 1304, F5, 25, 25) (dual of [(1304, 25), 32475, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(5125, 15649, F5, 25) (dual of [15649, 15524, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(5125, 15652, F5, 25) (dual of [15652, 15527, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(5121, 15626, F5, 25) (dual of [15626, 15505, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(597, 15626, F5, 21) (dual of [15626, 15529, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5125, 15652, F5, 25) (dual of [15652, 15527, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(5125, 15649, F5, 25) (dual of [15649, 15524, 26]-code), using
- net defined by OOA [i] based on linear OOA(5125, 1304, F5, 25, 25) (dual of [(1304, 25), 32475, 26]-NRT-code), using
(104, 104+25, 15661)-Net over F5 — Digital
Digital (104, 129, 15661)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5129, 15661, F5, 25) (dual of [15661, 15532, 26]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5126, 15655, F5, 25) (dual of [15655, 15529, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(5121, 15626, F5, 25) (dual of [15626, 15505, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(597, 15626, F5, 21) (dual of [15626, 15529, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(55, 29, F5, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(5126, 15658, F5, 24) (dual of [15658, 15532, 25]-code), using Gilbert–Varšamov bound and bm = 5126 > Vbs−1(k−1) = 8059 734121 036755 287755 417156 405299 618310 630872 118951 505682 086324 776670 208590 780426 487525 [i]
- linear OA(50, 3, F5, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(5126, 15655, F5, 25) (dual of [15655, 15529, 26]-code), using
- construction X with Varšamov bound [i] based on
(104, 104+25, large)-Net in Base 5 — Upper bound on s
There is no (104, 129, large)-net in base 5, because
- 23 times m-reduction [i] would yield (104, 106, large)-net in base 5, but