Best Known (81, 81+22, s)-Nets in Base 5
(81, 81+22, 1421)-Net over F5 — Constructive and digital
Digital (81, 103, 1421)-net over F5, using
- net defined by OOA [i] based on linear OOA(5103, 1421, F5, 22, 22) (dual of [(1421, 22), 31159, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(5103, 15631, F5, 22) (dual of [15631, 15528, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- 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(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(50, 6, F5, 0) (dual of [6, 6, 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
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- OA 11-folding and stacking [i] based on linear OA(5103, 15631, F5, 22) (dual of [15631, 15528, 23]-code), using
(81, 81+22, 7815)-Net over F5 — Digital
Digital (81, 103, 7815)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5103, 7815, F5, 2, 22) (dual of [(7815, 2), 15527, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5103, 15630, F5, 22) (dual of [15630, 15527, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5103, 15631, F5, 22) (dual of [15631, 15528, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- 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(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(50, 6, F5, 0) (dual of [6, 6, 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
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(5103, 15631, F5, 22) (dual of [15631, 15528, 23]-code), using
- OOA 2-folding [i] based on linear OA(5103, 15630, F5, 22) (dual of [15630, 15527, 23]-code), using
(81, 81+22, 4303822)-Net in Base 5 — Upper bound on s
There is no (81, 103, 4303823)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 986077 870515 594466 857910 820017 926322 410239 170836 145467 805050 011817 608213 > 5103 [i]