Automorphism groups and new constructions of maximum additive rank metric codes with restrictions

G. Longobardi; G. Lunardon; R. Trombetti; Y. Zhou

REVIEW 3 major objections 5 minor 19 references

Automorphism groups and new constructions of maximum additive rank metric codes with restrictions

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper shows the four known optimal restricted rank-metric code families are exactly intersections with shifted Gabidulin codes, yielding their automorphism groups, the equivalence criterion, and a new symmetric 2-code.

desk verdict Strong automorphism-group results for restricted MRD codes, but the non-equivalence proof for the new symmetric 2-code rests on an invalid WLOG step. read the letter →

arxiv 1908.02169 v1 pith:GI55CBFL submitted 2019-08-06 math.CO

G. Longobardi , G. Lunardon , R. Trombetti , Y. Zhou This is my paper · ORCID

classification math.CO MSC 94B0511T71

keywords rankmetriccodesMRDgeneralizedGabidulinsymmetricbilinearformsalternatingHermitianautomorphismgroupscodeequivalence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies maximum additive rank-metric codes whose matrices are required to be symmetric, alternating, or Hermitian. It establishes that the four known families of such codes are not isolated examples: each family is exactly the intersection of its restricted matrix space with a suitably shifted generalized Gabidulin code, and the shifted Gabidulin space is uniquely determined. From that intersection description it derives the full automorphism group of each family and proves that two codes are equivalent precisely when their defining shift exponents satisfy the same congruence up to sign. It also constructs a new maximum symmetric 2-code in even dimension with the same size as the known family but provably not equivalent to it. The result is a complete equivalence classification of the known restricted optimal codes and evidence that the list of such codes is not yet complete.

What carries the argument

The load-bearing object is the generalized Gabidulin code $G_{n,k,s}$, the set of linearized polynomials with q-degrees confined to $0,s,2s,\dots,(k-1)s$, together with its right-composition shifts $G\circ x^{q^{st}}$. The paper's key identity realizes each restricted code as $G'\cap X_n$, where $X_n$ is the subspace of self-adjoint symmetric, alternating, or Hermitian maps. What makes the identity powerful is Theorem 3.3: among all equivalent copies of the ambient Gabidulin code, the shifted container $G'$ is the unique one meeting $X_n$ in the given restricted code. Combined with the known classification of rank-preserving maps on these restricted spaces (all isometries have the form $\Psi_{a,g,\rho,r_0}(f)=a\,g\circ f^\rho\circ g^\top+r_0$, with an analogous $\Theta$ in the Hermitian case) and with the known automorphism group of $G_{n,k,s}$, the argument reduces automorphisms and equivalence of the restricted codes to monomial maps and to the congruence of $s$ modulo $n$.

What would settle it

The weakest point is easy to probe: for the excluded small parameters ($q=2,n=3$ for the symmetric space, $n\le 3$ for the alternating space), exhaustively compute the automorphism group of $S_{n,d,s}$ or $A_{n,d,s}$ over all rank-preserving maps; a single extra map would falsify Corollary 3.4 and Theorem 3.5. For the new code, a direct search over all maps $\Psi_{a,g,\rho}$ for small $m,q$ sending the Section 5 code into $S_{2m,2,s}$ would settle Theorem 5.1.

Watch

Extended reading notes

Core claim

The paper's central discovery is the intersection identity of Proposition 3.1: in q-polynomial form, the four known restricted maximum codes can be written as $S_{n,d,s}=G_{n,n-d+1,s}\circ x^{q^{s(n+d)/2}}\cap S_n(q)$, $A_{n,d,s}=G_{n,n-d+1,s}\circ x^{q^{sd/2}}\cap A_n(q)$, $H_{n,d,s}=G_{n,n-d+1,s}\circ x^{q^{s(n+d+1)}}\cap H_n(q^2)$, and $E_{n,d,s}=G_{n,n-d+1,s}\circ x^{q^{s(d+1)}}\cap H_n(q^2)$, where $G_{n,k,s}$ is the generalized Gabidulin code with minimum distance $d=n-k+1$. Theorem 3.3 proves that the shifted Gabidulin container is unique, and Theorem 4.1 characterizes every code equivalent to these families as the unique self-adjoint intersection $D=V\cap X_n$ with $V=V^\top$. From this the paper derives the automorphism groups in Corollary 3.4, proves the equivalence criterion $s\equiv\pm s'\pmod n$ in Theorem 3.5, and exhibits in Theorem 5.1 a maximum symmetric 2-code that is not equivalent to $S_{2m,2,s}$.

Load-bearing premise

The load-bearing premise is the full classification of rank-preserving maps on the restricted symmetric, alternating, and Hermitian matrix spaces; if that classification misses exceptional cases, then the automorphism groups and the equivalence criterion would have extra cases beyond the ones stated.

Editorial extensions

If this is right

Each of the four known families has exactly the stated monomial automorphism group; no rank-preserving map outside the displayed list fixes the code.
Within each family, the congruence $s\equiv\pm s'\pmod n$ is a complete invariant for equivalence: codes built from $s$ and $-s$ are equivalent, while codes built from inequivalent residues are not.
Every maximum code equivalent to one of these families is the intersection of its restricted space with a uniquely determined self-adjoint shifted Gabidulin space, so the whole equivalence class has the same structural description.
The new Section 5 code is a maximum symmetric 2-code with the same parameters as $S_{2m,2,s}$ but not equivalent to it, so the known symmetric family does not exhaust all maximum symmetric 2-codes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the uniqueness characterization extends to the punctured symmetric codes $T_{n,d,s}(\eta)$, those codes should also admit a Gabidulin-container description; checking that would give a direct route to their automorphism groups.
The same coefficient-comparison technique used against the new symmetric 2-code could be used as a test for equivalence between that code and any future symmetric 2-code family.
A natural classification question left implicit by the paper is whether the maximum symmetric 2-codes in $S_{2m}(q)$ consist of exactly two inequivalent families; the two examples here make that a concrete finite search.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear Colleague,

The paper is worth your attention: the automorphism-group and equivalence results for the four restricted MRD families are genuinely new and mostly well argued, but the advertised new symmetric 2-code in Section 5 is not yet proved to be new.

What the paper does well. Proposition 3.1 gives a clean characterization: each restricted maximum code is the intersection of its ambient symmetric/alternating/Hermitian space with a suitable shifted Gabidulin code. Theorem 3.3 shows that this intersection is unique, and the consequences—the automorphism groups in Corollary 3.4 and the equivalence criterion s ≡ ±s′ (mod n) in Theorem 3.5—are strong and appear correct for the symmetric and alternating cases, which are proved in detail. The Hermitian versions are only sketched, but the pattern is clear. Theorem 4.1's characterization of equivalence via a unique self-adjoint ambient subspace is also a nice contribution.

The soft spots. The proof of Theorem 5.1, which claims the new code S is not equivalent to S_{2m,2,s}, contains an invalid 'without loss of generality' step. From the coefficient vanishing the authors get g_{2m-i} g_{m-i}=0 for each i=1,...,m, but this only says that for each residue class modulo m at most one coefficient from the pair can be nonzero. It does not force all upper-half or all lower-half coefficients to vanish. Mixed patterns are possible, and some mixed patterns can even give permutation polynomials. The subsequent argument using c ∈ F_{q^m} depends on assuming the upper half is zero, so the non-equivalence proof does not go through as written. This is a load-bearing gap: the new-code claim is the headline of Section 5.

A second, smaller issue: the authors rely on Wan's classification of rank-preserving maps, which has small exceptional parameters, but the theorem statements do not exclude them. For example, q=2, n=3 for the symmetric case and n≤3 for alternating. This is likely fixable, but as stated the theorems are overbroad.

Bottom line. The intersection characterization and automorphism-group results are substantial and probably correct. The new 2-code may well be new, but the proof given is incomplete. The paper deserves a serious referee; I'd send it out with a specific request to verify Theorem 5.1. If that claim is repaired or removed, the rest is publishable.

Referee Report

3 major / 5 minor

Summary. The paper studies additive maximum rank-metric codes with symmetry restrictions: symmetric, alternating, and Hermitian forms. It gives a representation of the four known families (S_{n,d,s}, A_{n,d,s}, H_{n,d,s}, E_{n,d,s}) as intersections of shifted generalized Gabidulin codes with the ambient restricted space, determines their automorphism groups, proves an equivalence criterion under which two such codes are equivalent precisely when the defining parameter s is congruent to ±s' modulo n, and characterizes the known families by a uniqueness property. It then constructs a new maximum symmetric 2-code S in S_{2m}(q) and claims, via Theorem 5.1, that this code is not equivalent to Schmidt's code S_{2m,2,s}.

Significance. If the results are correct, the paper provides a clean structural description of the main known restricted MRD families and settles their equivalence and automorphism classification; the new code in Section 5 would be a genuinely new object in the symmetric setting. The intersection characterization and the explicit automorphism groups are potentially useful tools for further constructions. The main novelty, however, rests on the non-equivalence proof for S, and that proof contains a load-bearing gap; the Hermitian halves of three central results are also not proved in the text. The paper is therefore interesting but needs substantial revision before its central claims are supported.

major comments (3)

[§5, Theorem 5.1] The step 'Without loss of generality, we can suppose that g_{2m-i}=0 for i=1,...,m' is unjustified. The preceding coefficient identity gives only g_{2m-i}g_{m-i}=0 for each i, i.e. at least one factor in each antipodal pair vanishes. Mixed zero patterns are allowed. For example, when m=2, the conditions are g_3g_1=0 and g_2g_0=0; choosing g_1=g_2=0 and g_0,g_3 nonzero satisfies them, and g(x)=g_0x+g_3x^{q^{3s}} can be a permutation polynomial for suitable g_3. In such a pattern neither all upper-half coefficients nor all lower-half coefficients vanish, so the later argument that uses c∈F_{q^m} and concludes g_i=0 for i=0,...,m-1 does not apply. Since Theorem 5.1 is the only evidence that the constructed code S is new, this gap is load-bearing for the paper's main claim.
[§3, Theorem 3.3(ii); §3, Theorem 3.5; §4, Theorem 4.2] The Hermitian cases are not proved. Theorem 3.3(ii) is dismissed with 'the proof of this point is similar to that of previous one. For this reason we omit here the computations' and Theorem 4.2 with the same phrase. The Hermitian half of Theorem 3.5 is also delegated to 'similar arguments'. Since these theorems are stated as full classifications for H_{n,d,s} and E_{n,d,s}, the analogous coefficient arguments should be supplied or at least reduced in detail to the symmetric case; as written, the Hermitian claims are unsupported.
[§2 (after Eq. (9)) and §3, Corollary 3.4 / Theorem 3.5] The paper invokes Wan's classification of rank-preserving maps with stated exceptions (q=2,n=3 for S_n(q); n≤3 for A_n(q)), but the subsequent theorem statements do not exclude those parameters. In particular, Corollary 3.4 and Theorem 3.5 cover A_{n,d,s} for all n with 1<d<n, including n=3 and d=2, where the exceptional isometries of A_n(q) could alter the automorphism group or create additional equivalences. The authors should either exclude the exceptional parameters explicitly or verify that the theorems remain true in those cases.

minor comments (5)

[§5, first paragraph] The condition on s is printed as 'gcd(s,2m)' without the required equality '=1'; please correct.
[§3, Theorem 3.5] In the Hermitian part, the hypotheses use gcd(s,2n)=gcd(s',2n)=1 but the conclusion is s≡±s' (mod n). Since the Hermitian codes depend on s only modulo n, this is consistent, but the mismatch between the modulus in the hypothesis and the conclusion should be clarified.
[§5, paragraph before Theorem 5.1] The sentence 'the coefficients of terms x and x^{q^{s(2m-1)}} of f_m are c and ηb' uses an undefined symbol c; it should presumably be a (or another named coefficient).
[§3, proof of Theorem 3.3] The index conventions in the displayed formula for c_{m,j}(α_j) are hard to reconcile with the composition g∘αx^{q^{sj}}∘h; the power of q on h_{m-i-j} appears to be q^{s i} rather than q^{s(i+j)}. Please check the displayed formula and the resulting condition (23), and spell out the index arithmetic that yields the set equality after (23).
[§2, Theorem 2.3] The displayed bound for alternating codes is ambiguous: 'q^{n(n−1) 2m (m−e+1)}' should be written with the division by 2m clearly indicated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are proven by direct coefficient arguments, with self-citations [8] and [18] used only as external lemmas, not as the target conclusions.

full rationale

The derivation chain is self-contained once the quoted external classification and MRD-family results are accepted. Theorem 3.3 is proved by direct coefficient comparison: assuming W is equivalent to a generalized Gabidulin code and W ∩ Sn(q) = Sn,d,s forces the multipliers g and h to be monomials via the explicit zero-coefficient equations (23), with no fitted parameter or renamed input. Corollary 3.4 and Theorem 3.5 then combine this with Wan's external classification [19] and Sheekey's automorphism theorem [17]; the only self-citations, [8] and [18], are used as parameter-free external lemmas (equivalence of generalized Gabidulin codes and existence of the D family), not as assumptions of the restricted-code equivalence or newness being proved. Section 5's construction uses [18] only as an ingredient, and the claimed non-equivalence in Theorem 5.1 is attempted by a direct rank-preserving-map argument rather than by assuming the conclusion. The proof does contain a potential non-circularity gap, namely the 'Without loss of generality' step after g_{2m-i} g_{m-i}=0 in Theorem 5.1, but that is a correctness concern about product constraints, not a reduction of a prediction to an input. Overall, no claim in the paper is equivalent to its own input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper's conclusions rest on six external inputs: the Wan isometry classification, the Delsarte/Schmidt bounds, Sheekey's automorphism group of Gabidulin codes, the Lunardon-Trombetti-Zhou equivalence theorem for Gabidulin codes, the Trombetti-Zhou MRD code, and finite-field existence of nonsquare-norm elements. No free parameters were fitted to data, and no new entities were postulated.

assumptions (6)

domain assumption Complete classification of rank-distance preserving maps on X_n and H_n(q^2) from Wan [19]: every isometry is of the form Psi_{a,g,rho,r0} or Theta_{a,g,rho,r0}.
Invoked in Section 2 after Eq (9) and (15); used throughout Sections 3-5 to identify automorphism groups and equivalences. The paper itself notes exceptions (q=2,n=3 for S_n; n<=3 for A_n) but does not exclude them in later theorem statements.
domain assumption Singleton-like bound q^{n(n-d+1)} for unrestricted MRD codes (Delsarte) and restricted bounds of Schmidt (Thm 2.2), Delsarte-Goethals (Thm 2.3), Schmidt (Thm 2.4).
Used to call the constructions maximum and in Cor 3.4 to rule out a larger intersection D containing C.
domain assumption The automorphism group of the generalized Gabidulin code G_{n,k,s} is {Phi_{alpha x^{q^r}, id, beta x^{q^{n-r}}} : alpha,beta in F_{q^n}^*} (Sheekey [17]).
Used in the proof of Cor 3.4 to identify Aut(G') and in Theorem 3.3.
domain assumption For gcd(s,n)=gcd(s',n)=1, generalized Gabidulin codes G_s and G_s' are equivalent iff s is congruent to +/- s' modulo n (Lunardon-Trombetti-Zhou [8]).
Used in the forward direction of Theorem 3.5 to produce explicit equivalences between the restricted codes.
domain assumption The set D_{k,s}(eta) of Trombetti-Zhou [18] is an MRD code with minimum distance d=2m-k+1, and the rank lemma of Gow-Quinlan [5] used in Section 5.
Used in Section 5 to assert the new code S, obtained as an intersection with S_{2m}(q), has minimum distance 2.
standard math For odd q and m >= 2, there exists eta in F_{q^{2m}} with norm N_{q^{2m}/q}(eta) not a square.
Assumed at the start of Section 5 to define D_{k,s}(eta) and hence S; this is a standard existence fact for finite fields.

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Pith. "Pith review of Automorphism groups and new constructions of maximum additive rank metric codes with restrictions." pith.science (2026). https://pith.science/paper/GI55CBFL

@misc{pith2026190802169,
  author       = {Pith},
  title        = {Pith review of: Automorphism groups and new constructions of maximum additive rank metric codes with restrictions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GI55CBFL}},
  note         = {Machine review of arXiv:1908.02169}
}

abstract

Let $d, n \in \mathbb{Z}^+$ such that $1\leq d \leq n$. A $d$-code $\mathcal{C} \subset \mathbb{F}_q^{n \times n}$ is a subset of order $n$ square matrices with the property that for all pairs of distinct elements in $\mathcal{C}$, the rank of their difference is greater than or equal to $d$. A $d$-code with as many as possible elements is called a maximum $d$-code. The integer $d$ is also called the minimum distance of the code. When $d<n$, a classical example of such an object is the so-called generalized Gabidulin code. There exist several classes of maximum $d$-codes made up respectively of symmetric, alternating and hermitian matrices. In this article we focus on such examples. Precisely, we determine their automorphism groups and solve the equivalence issue for them. Finally, we exhibit a maximum symmetric $2$-code which is not equivalent to the one with same parameters known so far.

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

de la Cruz, M

J. de la Cruz, M. Kiermaier, A. W assermann, W. Willems: Al gebraic structures of MRD codes, Advances in Mathematics of Communications, vol. 10, p. 499-5 10, 2018

work page 2018
[2]

Delsarte, J.M

P. Delsarte, J.M. Goethals: Alternating bilinear forms over GF(q), Journal of Combinatorial Theory, Series A , vol. 19, p. 26-50, 1975

work page 1975
[3]

Delsarte: Bilinear forms over a ﬁnite ﬁeld, with appli cations to coding theory, Journal of Combinatorial Theory, Series A , vol

P. Delsarte: Bilinear forms over a ﬁnite ﬁeld, with appli cations to coding theory, Journal of Combinatorial Theory, Series A , vol. 25, Issue 3, p. 226-241, 1978

work page 1978
[4]

Gabidulin: Theory of codes with maximum rank distan ce, Problems of information transmission, vol 21, p

E.M. Gabidulin: Theory of codes with maximum rank distan ce, Problems of information transmission, vol 21, p. 3-16, 1985

work page 1985
[5]

R. Gow, R. Quinlan: Galois theory and linear algebra, Linear Algebra and its Applications , vol. 430, p. 1778-1789, 2009

work page 2009
[6]

Johnson, V

N.L. Johnson, V. Jha, M. Biliotti: Handbook of ﬁnite translation planes . Chapman & Hall/CRC, 2017

work page 2017
[7]

Gabidulin, A

E.M. Gabidulin, A. Kshevetskiy: The new construction of rank codes, Proceedings ISIT, 2005

work page 2005
[8]

Lunardon, R

G. Lunardon, R. Trombetti, Y. Zhou: Generalized twisted Gabidulin codes, Journal of Com- binatorial Theory, Series A , vol. 159, p. 79-106, 2018

work page 2018

Show all 19 references

[9]

Liebhold and G

D. Liebhold and G. Nebe: Automorphism groups of Gabiduli n-like codes. arXiv:1603.09565 [cs, math] , Mar. 2016

2016 arXiv
[10]

Lunardon: MRD-codes and linear sets, Journal of Combinatorial Theory, Series A , vol

G. Lunardon: MRD-codes and linear sets, Journal of Combinatorial Theory, Series A , vol. 149, p. 1-20, 2017

2017
[11]

Morrison: Equivalence for Rank-Metric codes and Aut omorphism Groups of Gabidulin Codes, IEEE Transection of Information Theory , vol

K. Morrison: Equivalence for Rank-Metric codes and Aut omorphism Groups of Gabidulin Codes, IEEE Transection of Information Theory , vol. 60, p. 7035-7046, 2014

2014
[12]

Lavrauw, O

M. Lavrauw, O. Polverino, Finite semiﬁelds . Chapter 6 in Current research topics in Galois Geometry (J. De Be Storme, Eds.), NOV A Academic Publishers, Pub. Date 2011, ISBN: 978-1-61209-523-3

2011
[13]

Schmidt: Hermitian rank distance codes, Designs, Codes and Cryptography , 86(7), p

K.U. Schmidt: Hermitian rank distance codes, Designs, Codes and Cryptography , 86(7), p. 1469–1481, 2018

2018
[14]

Schmidt: Quadratic and symmetric bilinear forms o ver ﬁnite ﬁelds and their association schemes, eprint arXiv:1803.04274 , 2018

K.U. Schmidt: Quadratic and symmetric bilinear forms o ver ﬁnite ﬁelds and their association schemes, eprint arXiv:1803.04274 , 2018

2018 arXiv
[15]

Schmidt: Symmetric bilinear forms over ﬁnite ﬁeld s of even characteristic, Journal of Combin

K.U. Schmidt: Symmetric bilinear forms over ﬁnite ﬁeld s of even characteristic, Journal of Combin. Theory Series A , vol. 117(8), p. 1011-1026, 2010

2010
[16]

Schmidt: Symmetric bilinear forms over ﬁnite ﬁeld s with applications to coding theory, Journal of Algebraic Combinatorics , vol

K.U. Schmidt: Symmetric bilinear forms over ﬁnite ﬁeld s with applications to coding theory, Journal of Algebraic Combinatorics , vol. 42, Issue 2, p. 635-670, 2015

2015
[17]

Sheekey: A new family of linear maximum rank distance codes, Advances in Mathematics of Communications , 2016

J. Sheekey: A new family of linear maximum rank distance codes, Advances in Mathematics of Communications , 2016

2016
[18]

Trombetti, Y

R. Trombetti, Y. Zhou: A new family of MRD codes in F2n q × F2n q with right and middle nuclei Fn q . IEEE Transection of Information Theory , to appear
[19]

W an: Geometry of matrices , W orld Scientiﬁc, Singapore, 1996

Z.X. W an: Geometry of matrices , W orld Scientiﬁc, Singapore, 1996. 18

1996

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

de la Cruz, M

J. de la Cruz, M. Kiermaier, A. W assermann, W. Willems: Al gebraic structures of MRD codes, Advances in Mathematics of Communications, vol. 10, p. 499-5 10, 2018

work page 2018

[2] [2]

Delsarte, J.M

P. Delsarte, J.M. Goethals: Alternating bilinear forms over GF(q), Journal of Combinatorial Theory, Series A , vol. 19, p. 26-50, 1975

work page 1975

[3] [3]

Delsarte: Bilinear forms over a ﬁnite ﬁeld, with appli cations to coding theory, Journal of Combinatorial Theory, Series A , vol

P. Delsarte: Bilinear forms over a ﬁnite ﬁeld, with appli cations to coding theory, Journal of Combinatorial Theory, Series A , vol. 25, Issue 3, p. 226-241, 1978

work page 1978

[4] [4]

Gabidulin: Theory of codes with maximum rank distan ce, Problems of information transmission, vol 21, p

E.M. Gabidulin: Theory of codes with maximum rank distan ce, Problems of information transmission, vol 21, p. 3-16, 1985

work page 1985

[5] [5]

R. Gow, R. Quinlan: Galois theory and linear algebra, Linear Algebra and its Applications , vol. 430, p. 1778-1789, 2009

work page 2009

[6] [6]

Johnson, V

N.L. Johnson, V. Jha, M. Biliotti: Handbook of ﬁnite translation planes . Chapman & Hall/CRC, 2017

work page 2017

[7] [7]

Gabidulin, A

E.M. Gabidulin, A. Kshevetskiy: The new construction of rank codes, Proceedings ISIT, 2005

work page 2005

[8] [8]

Lunardon, R

G. Lunardon, R. Trombetti, Y. Zhou: Generalized twisted Gabidulin codes, Journal of Com- binatorial Theory, Series A , vol. 159, p. 79-106, 2018

work page 2018