Best Known (21−11, 21, s)-Nets in Base 27
(21−11, 21, 146)-Net over F27 — Constructive and digital
Digital (10, 21, 146)-net over F27, using
- net defined by OOA [i] based on linear OOA(2721, 146, F27, 11, 11) (dual of [(146, 11), 1585, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2721, 731, F27, 11) (dual of [731, 710, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(2721, 729, F27, 11) (dual of [729, 708, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2719, 729, F27, 10) (dual of [729, 710, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(2721, 731, F27, 11) (dual of [731, 710, 12]-code), using
(21−11, 21, 150)-Net in Base 27 — Constructive
(10, 21, 150)-net in base 27, using
- 3 times m-reduction [i] based on (10, 24, 150)-net in base 27, using
- base change [i] based on digital (4, 18, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- base change [i] based on digital (4, 18, 150)-net over F81, using
(21−11, 21, 360)-Net over F27 — Digital
Digital (10, 21, 360)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2721, 360, F27, 2, 11) (dual of [(360, 2), 699, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2721, 366, F27, 2, 11) (dual of [(366, 2), 711, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2721, 732, F27, 11) (dual of [732, 711, 12]-code), using
- construction XX applied to C1 = C([727,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([727,9]) [i] based on
- linear OA(2719, 728, F27, 10) (dual of [728, 709, 11]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,8}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2719, 728, F27, 10) (dual of [728, 709, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2721, 728, F27, 11) (dual of [728, 707, 12]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2717, 728, F27, 9) (dual of [728, 711, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([727,9]) [i] based on
- OOA 2-folding [i] based on linear OA(2721, 732, F27, 11) (dual of [732, 711, 12]-code), using
- discarding factors / shortening the dual code based on linear OOA(2721, 366, F27, 2, 11) (dual of [(366, 2), 711, 12]-NRT-code), using
(21−11, 21, 53247)-Net in Base 27 — Upper bound on s
There is no (10, 21, 53248)-net in base 27, because
- 1 times m-reduction [i] would yield (10, 20, 53248)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 42392 731001 833326 905219 686401 > 2720 [i]