Best Known (11, 11+6, s)-Nets in Base 16
(11, 11+6, 1368)-Net over F16 — Constructive and digital
Digital (11, 17, 1368)-net over F16, using
- net defined by OOA [i] based on linear OOA(1617, 1368, F16, 6, 6) (dual of [(1368, 6), 8191, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(1617, 4104, F16, 6) (dual of [4104, 4087, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- 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(1610, 4096, F16, 4) (dual of [4096, 4086, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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 Ce(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(1617, 4104, F16, 6) (dual of [4104, 4087, 7]-code), using
(11, 11+6, 4130)-Net over F16 — Digital
Digital (11, 17, 4130)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1617, 4130, F16, 6) (dual of [4130, 4113, 7]-code), using
- 30 step Varšamov–Edel lengthening with (ri) = (1, 29 times 0) [i] based on linear OA(1616, 4099, F16, 6) (dual of [4099, 4083, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- 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(1613, 4096, F16, 5) (dual of [4096, 4083, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- 30 step Varšamov–Edel lengthening with (ri) = (1, 29 times 0) [i] based on linear OA(1616, 4099, F16, 6) (dual of [4099, 4083, 7]-code), using
(11, 11+6, 806563)-Net in Base 16 — Upper bound on s
There is no (11, 17, 806564)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 295 148762 670063 195631 > 1617 [i]