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{\it Review of:} Manuscript SNCS-D-19-00655
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{\it Journal:}  SN Computer Science
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{\it Title:}  A Dynamical System-based Key Equation for Decoding One-Point Algebraic Geometry Codes
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{\it Authors:}  R. Andriamifidisoa, R. Lalasoa, T. J. Rabeherimanana
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The authors develop a key equation for decoding one-point algebraic geometry 
codes, extending the approach in the well-known article by Heegard and Saints
(reference [7]).  The manuscript is clearly written and correct.  While 
the ideas involved are certainly attractive, I cannot recommend
that the manuscript be published. The problem is that essentially equivalent results have appeared 
a number of times previously in the literature, albeit in slightly different 
forms and making use of different language.  For instance, as far as the reviewer knows,
 the article ``Finite Linear Recurring Sequences and Homogeneous Ideals,'' A.A.E.C.C. 7 (1996), 
377--390, Althaler and D\"ur introduced the idea of writing key 
equations in the form the authors write as $G(X) \circ E = 0$ in Theorem 1, but
their paper interprets equations like this via Macaulay's inverse systems, so
that elements of the error-locator ideal are elements of the dual 
of the module generated by the syndrome series (this corresponds to 
the $E^\perp$ the authors introduce).  In their form,  
the syndrome series and the generators for the error-locator ideal are in
sets of variables written as $X_1,\ldots,X_r$ and the syndrome
series $E$ is written as a power series in $X_1^{-1},\ldots,X_r^{-1}$.
This interprets the operation written by the authors of the manuscript under review 
as $\circ$ is naturally via duality.  
An article by Little, ``The Key Equation for Codes from Order Domains,'' in: Advances 
in coding theory and cryptography, 1–16, Ser. Coding Theory Cryptol., 3, 
World Sci. Publ., Hackensack, NJ, 2007 extended Althaler and D\"ur's point of view to 
codes from order domains (which includes the one-point codes from 
curves considered by the authors, but which is also quite a bit more general).  
In the reviewer's opinion, the use of the ideas and language
from Oberst's presentation of linear systems (from reference [10]) do not add 
anything significantly new.  The ultimate result here
is just another form of a ``key equation'' for decoding equivalent to 
the forms described in the works listed above. Among the issues that would 
need to be resolved:
\item{1.}  The ``known'' syndromes from the received word usually do not determine the 
whole syndrome series so something like Feng-Rao majority voting is apparently 
still necessary.
\item{2.} Describing the decoding problem this way still leaves
the problem of computing what the authors call $G(X)$ (the Gr\"obner
basis of the error-locator ideal) from $E$ and the best way to do that
has not really be determined.
\item{3.}    This might be the 
first step towards improved decoding algorithms for these codes, but it is only a first step.
It has not yet been demonstrated that these attractive ideas really
add much to our knowledge regarding the decoding problem for these codes.
\bye
