Best Known (80, 80+20, s)-Nets in Base 5
(80, 80+20, 1564)-Net over F5 — Constructive and digital
Digital (80, 100, 1564)-net over F5, using
- 1 times m-reduction [i] based on digital (80, 101, 1564)-net over F5, using
- net defined by OOA [i] based on linear OOA(5101, 1564, F5, 21, 21) (dual of [(1564, 21), 32743, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5101, 15641, F5, 21) (dual of [15641, 15540, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5101, 15642, F5, 21) (dual of [15642, 15541, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- 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(585, 15626, F5, 17) (dual of [15626, 15541, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(54, 16, F5, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,5)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5101, 15642, F5, 21) (dual of [15642, 15541, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5101, 15641, F5, 21) (dual of [15641, 15540, 22]-code), using
- net defined by OOA [i] based on linear OOA(5101, 1564, F5, 21, 21) (dual of [(1564, 21), 32743, 22]-NRT-code), using
(80, 80+20, 13182)-Net over F5 — Digital
Digital (80, 100, 13182)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5100, 13182, F5, 20) (dual of [13182, 13082, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(5100, 15646, F5, 20) (dual of [15646, 15546, 21]-code), using
- 1 times truncation [i] based on linear OA(5101, 15647, F5, 21) (dual of [15647, 15546, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(597, 15625, F5, 21) (dual of [15625, 15528, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(20) ⊂ Ce(16) [i] based on
- 1 times truncation [i] based on linear OA(5101, 15647, F5, 21) (dual of [15647, 15546, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5100, 15646, F5, 20) (dual of [15646, 15546, 21]-code), using
(80, 80+20, large)-Net in Base 5 — Upper bound on s
There is no (80, 100, large)-net in base 5, because
- 18 times m-reduction [i] would yield (80, 82, large)-net in base 5, but