MinT - Best Known (123−20, 123, s)-Nets in Base 4

Best Known (123−20, 123, s)-Nets in Base 4

(123−20, 123, 6555)-Net over F₄ — Constructive and digital

Digital (103, 123, 6555)-net over F₄, using

1 times m-reduction [i] based on digital (103, 124, 6555)-net over F₄, using
- net defined by OOA [i] based on linear OOA(4¹²⁴, 6555, F₄, 21, 21) (dual of [(6555, 21), 137531, 22]-NRT-code), using
  - OOA 10-folding and stacking with additional row [i] based on linear OA(4¹²⁴, 65551, F₄, 21) (dual of [65551, 65427, 22]-code), using
    - discarding factors / shortening the dual code based on linear OA(4¹²⁴, 65555, F₄, 21) (dual of [65555, 65431, 22]-code), using
      - constructionÂ X applied to C_e(20) âŠ‚ C_e(17) [i] based on
        linear OA(4¹²¹, 65536, F₄, 21) (dual of [65536, 65415, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(4¹⁰⁵, 65536, F₄, 18) (dual of [65536, 65431, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
        linear OA(4³, 19, F₄, 2) (dual of [19, 16, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(4³, 21, F₄, 2) (dual of [21, 18, 3]-code), using
        Hamming code H(3,4) [i]

(123−20, 123, 32777)-Net over F₄ — Digital

Digital (103, 123, 32777)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4¹²³, 32777, F₄, 2, 20) (dual of [(32777, 2), 65431, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4¹²³, 65554, F₄, 20) (dual of [65554, 65431, 21]-code), using
  - discarding factors / shortening the dual code based on linear OA(4¹²³, 65555, F₄, 20) (dual of [65555, 65432, 21]-code), using
    - constructionÂ XX applied to C_e(20) âŠ‚ C_e(17) âŠ‚ C_e(16) [i] based on
      1. linear OA(4¹²¹, 65536, F₄, 21) (dual of [65536, 65415, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
      2. linear OA(4¹⁰⁵, 65536, F₄, 18) (dual of [65536, 65431, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
      3. linear OA(4⁹⁷, 65536, F₄, 17) (dual of [65536, 65439, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
      4. linear OA(4¹, 18, F₄, 1) (dual of [18, 17, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
      5. linear OA(4⁰, 1, F₄, 0) (dual of [1, 1, 1]-code), using
        dual of repetition code with length 1 [i]

(123−20, 123, large)-Net in Base 4 — Upper bound on s

There is no (103, 123, large)-net in base 4, because

18 times m-reduction [i] would yield (103, 105, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (123−20, 123, s)-Nets in Base 4

(123−20, 123, 6555)-Net over F4 — Constructive and digital

(123−20, 123, 32777)-Net over F4 — Digital

(123−20, 123, large)-Net in Base 4 — Upper bound on s

(123−20, 123, 6555)-Net over F₄ — Constructive and digital

(123−20, 123, 32777)-Net over F₄ — Digital