Best Known (51−11, 51, s)-Nets in Base 5
(51−11, 51, 3126)-Net over F5 — Constructive and digital
Digital (40, 51, 3126)-net over F5, using
- 51 times duplication [i] based on digital (39, 50, 3126)-net over F5, using
- net defined by OOA [i] based on linear OOA(550, 3126, F5, 11, 11) (dual of [(3126, 11), 34336, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(550, 15631, F5, 11) (dual of [15631, 15581, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(550, 15632, F5, 11) (dual of [15632, 15582, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(543, 15625, F5, 9) (dual of [15625, 15582, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(51, 7, F5, 1) (dual of [7, 6, 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 Ce(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(550, 15632, F5, 11) (dual of [15632, 15582, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(550, 15631, F5, 11) (dual of [15631, 15581, 12]-code), using
- net defined by OOA [i] based on linear OOA(550, 3126, F5, 11, 11) (dual of [(3126, 11), 34336, 12]-NRT-code), using
(51−11, 51, 7917)-Net over F5 — Digital
Digital (40, 51, 7917)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(551, 7917, F5, 11) (dual of [7917, 7866, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(551, 15634, F5, 11) (dual of [15634, 15583, 12]-code), using
- construction XX applied to Ce(10) ⊂ Ce(8) ⊂ Ce(7) [i] based on
- linear OA(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(543, 15625, F5, 9) (dual of [15625, 15582, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(537, 15625, F5, 8) (dual of [15625, 15588, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(51, 8, F5, 1) (dual of [8, 7, 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
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(10) ⊂ Ce(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(551, 15634, F5, 11) (dual of [15634, 15583, 12]-code), using
(51−11, 51, 6360277)-Net in Base 5 — Upper bound on s
There is no (40, 51, 6360278)-net in base 5, because
- 1 times m-reduction [i] would yield (40, 50, 6360278)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 88817 861624 978277 072233 460482 736345 > 550 [i]