Best Known (13, 13+4, s)-Nets in Base 5
(13, 13+4, 1568)-Net over F5 — Constructive and digital
Digital (13, 17, 1568)-net over F5, using
- net defined by OOA [i] based on linear OOA(517, 1568, F5, 4, 4) (dual of [(1568, 4), 6255, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(517, 1568, F5, 3, 4) (dual of [(1568, 3), 4687, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(517, 3136, F5, 4) (dual of [3136, 3119, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(1) [i] based on
- 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, 3125, F5, 2) (dual of [3125, 3119, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- linear OA(51, 11, F5, 1) (dual of [11, 10, 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(3) ⊂ Ce(1) [i] based on
- OA 2-folding and stacking [i] based on linear OA(517, 3136, F5, 4) (dual of [3136, 3119, 5]-code), using
- appending kth column [i] based on linear OOA(517, 1568, F5, 3, 4) (dual of [(1568, 3), 4687, 5]-NRT-code), using
(13, 13+4, 4447)-Net over F5 — Digital
Digital (13, 17, 4447)-net over F5, using
- net defined by OOA [i] based on linear OOA(517, 4447, F5, 4, 4) (dual of [(4447, 4), 17771, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(517, 4447, F5, 3, 4) (dual of [(4447, 3), 13324, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(517, 4447, F5, 4) (dual of [4447, 4430, 5]-code), using
- 1316 step Varšamov–Edel lengthening with (ri) = (1, 1315 times 0) [i] based on linear OA(516, 3130, F5, 4) (dual of [3130, 3114, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- 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(511, 3125, F5, 3) (dual of [3125, 3114, 4]-code or 3125-cap in PG(10,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 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(3) ⊂ Ce(2) [i] based on
- 1316 step Varšamov–Edel lengthening with (ri) = (1, 1315 times 0) [i] based on linear OA(516, 3130, F5, 4) (dual of [3130, 3114, 5]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(517, 4447, F5, 4) (dual of [4447, 4430, 5]-code), using
- appending kth column [i] based on linear OOA(517, 4447, F5, 3, 4) (dual of [(4447, 3), 13324, 5]-NRT-code), using
(13, 13+4, 308815)-Net in Base 5 — Upper bound on s
There is no (13, 17, 308816)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 762943 515905 > 517 [i]