Griesmer code[13] 


A \([n,k,d]_q\) code is a Griesmer code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the Griesmer bound \begin{align} n\geq\sum_{j=0}^{k-1}\left\lceil \frac{d}{q^{j}}\right\rceil ~, \tag*{(1)}\end{align} where \(\left\lceil x\right\rceil \) is the ceiling function, becomes an equality.

An \([n,2,d]_q\) code exists if and only if it is not excluded by the Griesmer bound. Every Griesmer code is generated by its minimum-weight codewords [4].




  • Divisible code — If a \(p\)-ary Griesmer code with \(p\) prime is such that a power of \(p\) divides the distance, then the code is divisible by that power [6].
  • Binary BCH code — The \([15,5,7]\) BCH code extended with a parity check saturates the Griesmer bound ([5], pg. 157).
  • Kasami code — Kasami codes satisfy the Griesmer bound for certain parameters [7].
  • \([7,4,3]\) Hamming code — Starting with the \([6,3,3]\) shortened Hamming code and applying the \((u|u+v)\) construction recursively using the repetition code yields a family of \([2^m,m+1,2^{m-1}]\) codes for \(m\geq1\) that saturate the Griesmer bound [5; pg. 90].
  • Anticode — Several anticode (e.g., [8,9]) and related [10] constructions saturate the Griesmer bound; see Refs. [5,11,12] for more details.
  • Projective RM (PRM) code — PRM codes for \(r=1\) attain the Griesmer bound for all \(m\) [13].
  • Grassmannian code — The binary Grassmannian \([35,6,16]\) code, whose points lie on the Grassmannian \({\mathbb{G}(2,4)}\), attains the Griesmer bound [13].
  • Projective geometry code — Acrs corresponding to Griesmer codes are called Griesmer arcs [14; pg. 248]. There is a one-to-one correspondence between Griesmer codes and minihypers [15,16]; see [18][17; Sec. 14.2.4] for more details.


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“Griesmer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_griesmer, title={Griesmer code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Griesmer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.