Best Known (124−25, 124, s)-Nets in Base 5
(124−25, 124, 1303)-Net over F5 — Constructive and digital
Digital (99, 124, 1303)-net over F5, using
- 52 times duplication [i] based on digital (97, 122, 1303)-net over F5, using
- net defined by OOA [i] based on linear OOA(5122, 1303, F5, 25, 25) (dual of [(1303, 25), 32453, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(5122, 15637, F5, 25) (dual of [15637, 15515, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(5122, 15639, F5, 25) (dual of [15639, 15517, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [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(5109, 15626, F5, 23) (dual of [15626, 15517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(51, 13, F5, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5122, 15639, F5, 25) (dual of [15639, 15517, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(5122, 15637, F5, 25) (dual of [15637, 15515, 26]-code), using
- net defined by OOA [i] based on linear OOA(5122, 1303, F5, 25, 25) (dual of [(1303, 25), 32453, 26]-NRT-code), using
(124−25, 124, 12877)-Net over F5 — Digital
Digital (99, 124, 12877)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5124, 12877, F5, 25) (dual of [12877, 12753, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(5124, 15646, F5, 25) (dual of [15646, 15522, 26]-code), using
- 1 times truncation [i] based on linear OA(5125, 15647, F5, 26) (dual of [15647, 15522, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(5121, 15625, F5, 26) (dual of [15625, 15504, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(5103, 15625, F5, 22) (dual of [15625, 15522, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(54, 22, F5, 3) (dual of [22, 18, 4]-code or 22-cap in PG(3,5)), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- 1 times truncation [i] based on linear OA(5125, 15647, F5, 26) (dual of [15647, 15522, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(5124, 15646, F5, 25) (dual of [15646, 15522, 26]-code), using
(124−25, 124, large)-Net in Base 5 — Upper bound on s
There is no (99, 124, large)-net in base 5, because
- 23 times m-reduction [i] would yield (99, 101, large)-net in base 5, but