Best Known (13, 13+7, s)-Nets in Base 16
(13, 13+7, 1368)-Net over F16 — Constructive and digital
Digital (13, 20, 1368)-net over F16, using
- net defined by OOA [i] based on linear OOA(1620, 1368, F16, 7, 7) (dual of [(1368, 7), 9556, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1620, 4105, F16, 7) (dual of [4105, 4085, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(1619, 4097, F16, 7) (dual of [4097, 4078, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(1613, 4097, F16, 5) (dual of [4097, 4084, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(167, 8, F16, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,16)), using
- dual of repetition code with length 8 [i]
- linear OA(161, 8, F16, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, 16, F16, 1) (dual of [16, 15, 2]-code), using
- Reed–Solomon code RS(15,16) [i]
- discarding factors / shortening the dual code based on linear OA(161, 16, F16, 1) (dual of [16, 15, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(1620, 4105, F16, 7) (dual of [4105, 4085, 8]-code), using
(13, 13+7, 4110)-Net over F16 — Digital
Digital (13, 20, 4110)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1620, 4110, F16, 7) (dual of [4110, 4090, 8]-code), using
- 10 step Varšamov–Edel lengthening with (ri) = (1, 9 times 0) [i] based on linear OA(1619, 4099, F16, 7) (dual of [4099, 4080, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(1619, 4096, F16, 7) (dual of [4096, 4077, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(1616, 4096, F16, 6) (dual of [4096, 4080, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(160, 3, F16, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- 10 step Varšamov–Edel lengthening with (ri) = (1, 9 times 0) [i] based on linear OA(1619, 4099, F16, 7) (dual of [4099, 4080, 8]-code), using
(13, 13+7, 5121363)-Net in Base 16 — Upper bound on s
There is no (13, 20, 5121364)-net in base 16, because
- 1 times m-reduction [i] would yield (13, 19, 5121364)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 75557 880183 768297 516631 > 1619 [i]