MinT - Best Known (17−3, 17, s)-Nets in Base 3

Best Known (17−3, 17, s)-Nets in Base 3

(17−3, 17, 307328)-Net over F₃ — Constructive and digital

Digital (14, 17, 307328)-net over F₃, using

net defined by OOA [i] based on linear OOA(3¹⁷, 307328, F₃, 3, 3) (dual of [(307328, 3), 921967, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(3¹⁷, 307328, F₃, 2, 3) (dual of [(307328, 2), 614639, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3¹⁷, 307328, F₃, 3) (dual of [307328, 307311, 4]-code or 307328-cap in PG(16,3)), using
    - product of two projective caps and 2-cap in AG(1,Â 3) [i] based on
      1. linear OA(3⁶, 56, F₃, 3) (dual of [56, 50, 4]-code or 56-cap in PG(5,3)), using
        the Hill cap [i]
      2. linear OA(3¹¹, 2744, F₃, 3) (dual of [2744, 2733, 4]-code or 2744-cap in PG(10,3)), using
        caps completed using a computer search [i]

(17−3, 17, 3693076)-Net over F₃ — Upper bound on s (digital)

There is no digital (14, 17, 3693077)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁷, 3693077, F₃, 3) (dual of [3693077, 3693060, 4]-code or 3693077-cap in PG(16,3)), but
- doubling the cap [i] would yield 7386154-cap in AG(17,3), but
  - 11 times the recursive bound from Bierbrauer and Edel [i] would yield 113-cap in AG(6,3), but
    - bound on the size of affine caps [i]

(17−3, 17, large)-Net in Base 3 — Upper bound on s

There is no (14, 17, large)-net in base 3, because

1 times m-reduction [i] would yield (14, 16, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]