Best Known (126, 126+23, s)-Nets in Base 5
(126, 126+23, 35513)-Net over F5 — Constructive and digital
Digital (126, 149, 35513)-net over F5, using
- 52 times duplication [i] based on digital (124, 147, 35513)-net over F5, using
- net defined by OOA [i] based on linear OOA(5147, 35513, F5, 23, 23) (dual of [(35513, 23), 816652, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5147, 390644, F5, 23) (dual of [390644, 390497, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(5146, 390643, F5, 23) (dual of [390643, 390497, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(5145, 390626, F5, 23) (dual of [390626, 390481, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 516−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(5129, 390626, F5, 21) (dual of [390626, 390497, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 516−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(51, 17, F5, 1) (dual of [17, 16, 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([0,11]) ⊂ C([0,10]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(5146, 390643, F5, 23) (dual of [390643, 390497, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5147, 390644, F5, 23) (dual of [390644, 390497, 24]-code), using
- net defined by OOA [i] based on linear OOA(5147, 35513, F5, 23, 23) (dual of [(35513, 23), 816652, 24]-NRT-code), using
(126, 126+23, 195325)-Net over F5 — Digital
Digital (126, 149, 195325)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5149, 195325, F5, 2, 23) (dual of [(195325, 2), 390501, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5149, 390650, F5, 23) (dual of [390650, 390501, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5149, 390651, F5, 23) (dual of [390651, 390502, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(5145, 390625, F5, 23) (dual of [390625, 390480, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(5121, 390625, F5, 19) (dual of [390625, 390504, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(5149, 390651, F5, 23) (dual of [390651, 390502, 24]-code), using
- OOA 2-folding [i] based on linear OA(5149, 390650, F5, 23) (dual of [390650, 390501, 24]-code), using
(126, 126+23, large)-Net in Base 5 — Upper bound on s
There is no (126, 149, large)-net in base 5, because
- 21 times m-reduction [i] would yield (126, 128, large)-net in base 5, but