Best Known (17, 17+6, s)-Nets in Base 5
(17, 17+6, 1044)-Net over F5 — Constructive and digital
Digital (17, 23, 1044)-net over F5, using
- 51 times duplication [i] based on digital (16, 22, 1044)-net over F5, using
- net defined by OOA [i] based on linear OOA(522, 1044, F5, 6, 6) (dual of [(1044, 6), 6242, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(522, 3132, F5, 6) (dual of [3132, 3110, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(521, 3125, F5, 6) (dual of [3125, 3104, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(516, 3125, F5, 4) (dual of [3125, 3109, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(56, 7, F5, 6) (dual of [7, 1, 7]-code or 7-arc in PG(5,5)), using
- dual of repetition code with length 7 [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 X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(522, 3132, F5, 6) (dual of [3132, 3110, 7]-code), using
- net defined by OOA [i] based on linear OOA(522, 1044, F5, 6, 6) (dual of [(1044, 6), 6242, 7]-NRT-code), using
(17, 17+6, 3135)-Net over F5 — Digital
Digital (17, 23, 3135)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(523, 3135, F5, 6) (dual of [3135, 3112, 7]-code), using
- 2 step Varšamov–Edel lengthening with (ri) = (1, 0) [i] based on linear OA(522, 3132, F5, 6) (dual of [3132, 3110, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(521, 3125, F5, 6) (dual of [3125, 3104, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(516, 3125, F5, 4) (dual of [3125, 3109, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(56, 7, F5, 6) (dual of [7, 1, 7]-code or 7-arc in PG(5,5)), using
- dual of repetition code with length 7 [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 X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- 2 step Varšamov–Edel lengthening with (ri) = (1, 0) [i] based on linear OA(522, 3132, F5, 6) (dual of [3132, 3110, 7]-code), using
(17, 17+6, 103773)-Net in Base 5 — Upper bound on s
There is no (17, 23, 103774)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 11921 100951 496089 > 523 [i]