On Extremal Self-Dual Ternary Codes of Length 48

Abstract

All extremal ternary self-dual codes of length 48 that have some automorphism of prime order 𝑝≥5 are equivalent to one of the two known codes, the Pless code or the extended quadratic residue code.

1. Introduction

The notion of an extremal self-dual code has been introduced in [1]. As Gleason [2] remarks one may use invariance properties of the weight enumerator of a self-dual code to deduce upper bounds on the minimum distance. Extremal codes are self-dual codes that achieve these bounds. The most wanted extremal code is a binary self-dual doubly even code of length 72 and minimum distance 16. One frequently used strategy is to classify extremal codes with a given automorphism, see [3, 4] for the first papers on this subject.

Ternary codes with a given automorphism have been studied in [5]. The minimum distance 𝑑(𝐶)∶=min{𝑤𝑡(𝑐)∣0≠𝑐∈𝐶} of a self-dual ternary code 𝐶=𝐶^⟂≤𝔽^𝑛₃ of length 𝑛 is bounded by𝑑(𝐶)𝑛≤312+3. (1.1) Codes achieving equality are called extremal. Of particular interest are extremal ternary codes of length a multiple of 12. There exists a unique extremal code of length 12 (the extended ternary Golay code), two extremal codes of length 24 (the extended quadratic residue code 𝑄₂₄∶=𝑄𝑅(23,3) and the Pless code 𝑃₂₄). For length 36, the Pless code yields one example of an extremal code. Reference [5] shows that this is the only code with an automorphism of prime order 𝑝≥5; a complete classification is yet unknown. The present paper investigates the extremal codes of length 48. There are two such codes known, the extended quadratic residue code 𝑄₄₈ and the Pless code 𝑃₄₈. The computer calculations described in this paper show that these two codes are the only extremal ternary codes 𝐶 of length 48 for which the order of the automorphism group is divisible by some prime 𝑝≥5. Theoretical arguments exclude all types of automorphisms that do not occur for the two known examples.

Any extremal ternary self-dual code of length 48 defines an extremal even unimodular lattice of dimension 48 ([6]). A long-term project to find or even classify such lattices was my main motivation for this paper.

2. Automorphisms of Codes

Let 𝔽 be some finite field, 𝔽^∗ its multiplicative group. For any monomial transformation 𝜎∈Mon_𝑛(𝔽)∶=𝔽^∗≀𝑆_𝑛, the image 𝜋(𝜎)∈𝑆_𝑛 is called the permutational part of 𝜎. Then 𝜎 has a unique expression as𝜎=diag𝛼₁,…,𝛼_𝑛𝜋(𝜎)=𝑚(𝜎)𝜋(𝜎), (2.1) and 𝑚(𝜎) is called the monomial part of 𝜎. For a code 𝐶≤𝔽^𝑛 we letMon(𝐶)∶=𝜎∈Mon_𝑛(𝔽)∣𝜎(𝐶)=𝐶 (2.2) be the full monomial automorphism group of 𝐶.

We call a code 𝐶≤𝔽^𝑛 an orthogonal direct sum, if there are codes 𝐶_𝑖≤𝔽^𝑛𝑖 (1≤𝑖≤𝑠>1) of length 𝑛_𝑖 such that𝐶∼0𝑥02^𝑠9𝑏9_𝑖=1𝐶_𝑖=𝑐₁⁽¹⁾,…,𝑐_𝑛⁽¹⁾₁,…,𝑐₁^(𝑠),…,𝑐_𝑛^(𝑠)_𝑠∣𝑐^(𝑖)∈𝐶_𝑖(1≤𝑖≤𝑠). (2.3)

Lemma 2.1. Let 𝐶≤𝔽^𝑛 not be an orthogonal direct sum. Then the kernel of the restriction of 𝜋 to Mon(𝐶) is isomorphic to 𝔽^∗.

Proof. Clearly 𝔽^∗𝐶=𝐶 since 𝐶 is an 𝔽-subspace. Assume that 𝜎∶=diag(𝛼₁,…,𝛼_𝑛)∈Mon(𝐶) with 𝛼_𝑖∈𝔽^∗, not all equal. Let {𝛼₁,…,𝛼_𝑛}={𝛽₁,…,𝛽_𝑠} with pairwise distinct 𝛽_𝑖. Then 𝐶=0𝑥02^𝑠9𝑏9_𝑖=1ker𝜎−𝛽_𝑖id (2.4) is the direct sum of eigenspaces of 𝜎. Moreover the standard basis is a basis of eigenvectors of 𝜎 so this is an orthogonal direct sum.

In the investigation of possible automorphisms of codes, the following strategy has proved to be very fruitful ([4, 7]).

Definition 2.2. Let 𝜎∈Mon(𝐶) be an automorphism of 𝐶. Then 𝜋(𝜎)∈𝑆_𝑛 is a direct product of disjoint cycles of lengths dividing the order of 𝜎. In particular if the order of 𝜎 is some prime 𝑝, then we say that 𝜎 has cycle type (𝑡,𝑓), if 𝜋(𝜎) has 𝑡 cycles of length 𝑝 and 𝑓 fixed points (so 𝑝𝑡+𝑓=𝑛).

Lemma 2.3. Let 𝜎∈Mon(𝐶) have prime order 𝑝.(a)If 𝑝 does not divide |𝔽^∗| then there is some element 𝜏∈Mon_𝑛(𝔽) such that 𝑚(𝜏𝜎𝜏⁻¹)=𝑖𝑑. Replacing 𝐶 by 𝜏(𝐶) one hence may assume that 𝑚(𝜎)=1.(b)Assume that 𝑝 does not divide char(𝔽), 𝑚(𝜎)=1, and 𝜋(𝜎)=(1,…,𝑝)⋯((𝑡−1)𝑝+1,…,𝑡𝑝)(𝑡𝑝+1)⋯(𝑛). Then 𝐶=𝐶(𝜎)⊕𝐸, where 𝐶(𝜎)=𝑐∈𝐶∣𝑐₁=⋯=𝑐_𝑝,𝑐_𝑝+1=⋯=𝑐_2𝑝,…,𝑐_{(𝑡−1)𝑝+1}=⋯=𝑐_𝑡𝑝 (2.5) is the fixed code of 𝜎 and 𝐸=𝑐∈𝐶∣^𝑝_𝑖=1𝑐_𝑖=^2𝑝_𝑖=𝑝+1𝑐_𝑖=⋯=^𝑡𝑝_{()𝑖=𝑡−1𝑝+1}𝑐_𝑖=𝑐_𝑡𝑝+1=⋯=𝑐_𝑛=0 (2.6) is the unique 𝜎-invariant complement of 𝐶(𝜎) in 𝐶.(c)Define two projections 𝜋_𝑡∶𝐶(𝜎)→𝔽^𝑡,𝜋_𝑡(𝑐)∶=𝑐_𝑝,𝑐_2𝑝,…,𝑐_𝑡𝑝,𝜋_𝑓∶𝐶(𝜎)→𝔽^𝑓,𝜋_𝑓(𝑐)∶=𝑐_𝑡𝑝+1,𝑐_𝑡𝑝+2,…,𝑐_{𝑡𝑝+𝑓}. (2.7) So 𝐶(𝜎)≅(𝜋_𝑡(𝐶(𝜎)),𝜋_𝑓(𝐶(𝜎)))=∶𝐶(𝜎)^∗. If 𝐶=𝐶^⟂ is self-dual with respect to (𝑥,𝑦)∶=∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖, then 𝐶(𝜎)^∗≤𝔽^𝑡+𝑓 is a self-dual code with respect to the inner product (𝑥,𝑦)∶=∑^𝑡_𝑖=1𝑝𝑥_𝑖𝑦_𝑖+∑^𝑡+𝑓_𝑗=𝑡+1𝑥_𝑗𝑦_𝑗.(d)In particular dim(𝐶(𝜎))=(𝑡+𝑓)/2 and dim(𝐸)=𝑡(𝑝−1)/2.

Proof. (a) follows from the Schur-Zassenhaus theorem in finite group theory. For the ternary case, see [5, Lemma  1].
(b) and (c) are similar to [4, Lemma  2].

In the following we will keep the notation of the previous lemma and regard the fixed code 𝐶(𝜎).

Remark 2.4. If 𝑓≤𝑑(𝐶) then 𝑡≥𝑓.

Proof. Otherwise the kernel 𝐾∶=ker(𝜋_𝑡)={(0,…,0,𝑐₁,…,𝑐_𝑓)∈𝐶(𝜎)} is a nontrivial subcode of minimum distance ≤𝑓<𝑑(𝐶).

The way to analyse the code 𝐸 from Lemma 2.3 is based on the following remark.

Remark 2.5. Let 𝑝≠char(𝔽) be some prime and 𝜎∈Mon_𝑛(𝔽) an element of order 𝑝. Let 𝑋^𝑝−1=(𝑋−1)𝑔₁⋯𝑔_𝑚∈𝔽[𝑋] (2.8) be the factorization of 𝑋^𝑝−1 into irreducible polynomials. Then all factors 𝑔_𝑖 have the same degree 𝑑=|⟨|𝔽|+𝑝ℤ⟩|, the order of |𝔽| mod 𝑝. There are polynomials 𝑎_𝑖∈𝔽[𝑋] (0≤𝑖≤𝑚) such that 1=𝑎₀𝑔₁⋯𝑔_𝑚+(𝑋−1)^𝑚_𝑖=1𝑎_{𝑖𝑗≠𝑖}𝑔_𝑗. (2.9) Then the primitive idempotents in 𝔽[𝑋]/(𝑋^𝑝−1) are given by the classes of 𝑒₀̃=𝑎₀𝑔₁⋯𝑔_𝑚,𝑒_𝑖̃=𝑎_{𝑖𝑗≠𝑖}𝑔_𝑗(𝑋−1),1≤𝑖≤𝑚. (2.10) Let 𝐿 be the extension field of 𝔽 with [𝐿∶𝔽]=𝑑. Then the group ring 𝔽[𝑋]𝑋^𝑝⟨𝜎⟩−1=𝔽≅𝔽⊕𝐿⊕⋯⊕𝐿_𝑚 (2.11) is a commutative semisimple 𝔽-algebra. Any code 𝐶≤𝔽^𝑛 with an automorphism 𝜎∈Mon(𝐶) is a module for this algebra. Put 𝑒_𝑖∶=𝑒_𝑖̃(𝜎)∈𝔽[𝜎]. Then 𝐶=𝐶𝑒₀⊕𝐶𝑒₁⊕⋯⊕𝐶𝑒_𝑚 with 𝐶𝑒₀=𝐶(𝜎), 𝐸=𝐶𝑒₁⊕⋯⊕𝐶𝑒_𝑚. Omitting the coordinates of 𝐸 that correspond to the fixed points of 𝜎, the codes 𝐶𝑒_𝑖 are 𝐿-linear codes of length 𝑡. Clearly dim_𝔽(𝐸)=𝑑∑^𝑚_𝑖=1dim_𝐿(𝐶𝑒_𝑖). If 𝐶 is self-dual then dim(𝐸)=𝑡(𝑝−1)/2.

3. Extremal Ternary Codes of Length 48

Let 𝐶=𝐶^⟂≤𝔽₃⁴⁸ be an extremal self-dual ternary code of length 48, so 𝑑(𝐶)=15.

3.1. Large Primes

In this section we prove the main result of this paper.

Theorem 3.1. Let 𝐶=𝐶^⟂≤𝔽₃⁴⁸ be an extremal self-dual code with an automorphism of prime order 𝑝≥5. Then 𝐶 is one of the two known codes. So either 𝐶=𝑄₄₈ is the extended quadratic residue code of length 48 with automorphism group Mon𝑄₄₈=𝐶₂×𝑃𝑆𝐿₂(47)oforder2⁵⋅3⋅23⋅47 (3.1) or 𝐶=𝑃₄₈ is the Pless code with automorphism group Mon𝑃₄₈=𝐶₂×𝑆𝐿₂(23)⋅2oforder2⁶⋅3⋅11⋅23. (3.2)

Lemma 3.2. Let 𝜎∈Mon(𝐶) be an automorphism of prime order 𝑝≥5. Then either 𝑝=47 and (𝑡,𝑓)=(1,1) or 𝑝=23 and (𝑡,𝑓)=(2,2) or 𝑝=11 and (𝑡,𝑓)=(4,4).

Proof. For the proof we use the notation of Lemma 2.3. In particular we let 𝐾∶=ker(𝜋_𝑡)={(0,…,0,𝑐₁,…,𝑐_𝑓)∈𝐶(𝜎)} and put 𝐾^∗∶={(𝑐₁,…,𝑐_𝑓)∣(0,…,0,𝑐₁,…,𝑐_𝑓)∈𝐶(𝜎)}. Then 𝐾^∗≤𝔽^𝑓₃,𝑑𝐾^∗≥15,dim𝐾^∗≥𝑓−𝑡2. (3.3) Moreover 𝑡𝑝+𝑓=48.
(1) If 𝑡=1, then 𝑝=47. If 𝑝=47, then 𝑡=𝑓=1. So assume that 𝑝<47 and 𝑡=1. Then the code 𝐸 has length 𝑝 and dimension (𝑝−1)/2, therefore 𝑝≥𝑑(𝐶)=15. So 𝑝≥17 and 𝑓≤48−17=31.
Then 𝐾^∗≤𝔽^𝑓₃ has dimension (𝑓−1)/2 and minimum distance 𝑑(𝐾^∗)≥15. From the bounds given in [8] there is no such possibility for 𝑓≤31.
(2) If 𝑡=2, then 𝑝=23. Assume that 𝑡=2. Since 2⋅𝑝≤48 we get 𝑝≤23, and if 𝑝=23, then (𝑡,𝑓)=(2,2).
So assume that 𝑝<23. The code 𝐸 is a nonzero code of length 2𝑝 and minimum distance ≥15, so 2𝑝≥15 and 𝑝 is one of 11, 13, 17, 19, and 𝑓=26, 22, 14, 10. The code 𝐾^∗≤𝔽^𝑓₃ has dimension ≥𝑓/2−1 and minimum distance ≥15. Again by [8] there is no such code.
(3) 𝑝≠13. For 𝑝=13 one now only has the possibility 𝑡=3 and 𝑓=9. The same argument as above constructs a code 𝐾^∗≤𝔽⁹₃ of dimension at least (𝑓+𝑡)/2−𝑡=3 of minimum distance ≥15>𝑓 which is absurd.
(4) If 𝑝=11, then 𝑡=𝑓=4. Otherwise 𝑡=3 and 𝑓=15 and the code 𝐾^∗ as above has length 15, dimension ≥6, and minimum distance ≥15 which is impossible.
(5) If 𝑝=7, then 𝑡=𝑓=6. Otherwise 𝑡=3,4,5 and 𝑓=27, 20, 13 and the code 𝐾^∗ as above has dimension ≥(𝑓+𝑡)/2−𝑡=12,8,4, length 𝑓, and minimum distance ≥15 which is impossible by [8].
(6) 𝑝≠7. Assume that 𝑝=7, then 𝑡=𝑓=6 and the kernel 𝐾 of the projection of 𝐶(𝜎) onto the first 42 components is trivial. So the image of the projection is 𝔽⁶₃⊗⟨(1,1,1,1,1,1,1)⟩; in particular it contains the vector (1⁷,0³⁵) of weight 7. So 𝐶(𝜎) contains some word (1⁷,0³⁵,𝑎₁,…,𝑎₆) of weight ≤13 which is a contradiction.
(7) If 𝑝=5, then 𝑡=𝑓=8 or 𝑡=9and𝑓=3. Otherwise 𝑡=3,4,5,6,7 and 𝑓=33,28,23,18,13 and the code 𝐾^∗≤𝔽^𝑓₃ has dimension ≥(𝑓+𝑡)/2−𝑡=15,12,9,6,3 and minimum distance ≥15 which is impossible by [8].
(8) 𝑝≠5. Assume that 𝑝=5. Then one possibility is that 𝑡=8 and the projection of 𝐶(𝜎) onto the first 8⋅5 coordinates is 𝔽⁸₃⊗⟨(1,1,1,1,1)⟩ and contains a word of weight 5. But then 𝐶(𝜎) has a word of weight 𝑤 with 5<𝑤≤5+8=13 a contradiction.
The other possibility is 𝑡=9. Then the code 𝐸=𝐸^⟂ is a Hermitian self-dual code of length 9 over the field with 3⁴=81 elements, which is impossible, since the length of such a code is 2 times the dimension and hence even.

Lemma 3.3. If 𝑝=11, then 𝐶≅𝑃₄₈.

Proof. Let 𝜎∈Mon(𝐶) be of order 11. Since (𝑥¹¹−1)=(𝑥−1)𝑔ℎ∈𝔽₃[𝑥] for irreducible polynomials 𝑔,ℎ of degree 5, 𝔽₃⟨𝜎⟩≅𝔽₃⊕𝔽₃⁵⊕𝔽₃⁵. (3.4) Let 𝑒₁,𝑒₂,𝑒₃∈𝔽₃⟨𝜎⟩ denote the primitive idempotents. Then 𝐶=𝐶𝑒₁⊕𝐶𝑒₂⊕𝐶𝑒₃ with 𝐶(𝜎)=𝐶𝑒₁=𝐶𝑒^⟂₁ of dimension 4 and 𝐶𝑒₂=𝐶𝑒^⟂₃≤(𝔽₃⁵⊕𝔽₃⁵)⁴. Clearly the projection of 𝐶(𝜎) onto the first 44 coordinates is injective. Since all weights of 𝐶 are multiples of 3 and ≥15, this leaves just one possibility for 𝐶(𝜎): ()1𝐺0=𝐿0∣𝑅0∶=¹¹0¹¹0¹¹0¹¹01111₁₁1¹¹0¹¹0¹¹011−1−1₁₁0¹¹1¹¹0¹¹01−11−1₁₁0¹¹0¹¹1¹¹1−1−11. (3.5) The cyclic code 𝑍 of length 11 with generator polynomial (𝑥−1)𝑔 (and similarly the one with generator polynomial (𝑥−1)ℎ) has weight enumerator 𝑥¹¹+132𝑥⁵𝑦⁶+110𝑥²𝑦⁹. (3.6) In particular it contains more words of weight 6 than of weight 9. This shows that the dimension of 𝐶𝑒_𝑖 over 𝔽₃⁵ is 2 for both 𝑖=2,3, since otherwise one of them has dimension ≥3 and therefore contains all words (0,0,𝑐,𝛼𝑐) for all 𝑐∈𝑍 and some 𝛼∈𝔽₃⁵. Not all of them can have weight ≥15. Similarly one sees that the codes 𝐶𝑒_𝑖≤𝔽⁴₃₅ have minimum distance 3 for 𝑖=2,3. So we may choose generator matrices 𝐺1∶=10𝑎𝑏01𝑐𝑑,𝐺2∶=10𝑎^′𝑏^′01𝑐^′𝑑^′ (3.7) with ^𝑎𝑏_𝑐𝑑∈𝐺𝐿₂(𝔽₃⁵) and ^{𝑎′𝑏′}_{𝑐′𝑑′}=−^𝑎𝑏_𝑐𝑑^−𝑡𝑟. To obtain 𝔽₃-generator matrices for the corresponding codes 𝐶𝑒₂ and 𝐶𝑒₃ of length 48, we choose a generator matrix 𝑔₁∈𝔽₃^5×11 of the cyclic code 𝑍 of length 11 with generator polynomial (𝑥−1)𝑔 and the corresponding dual basis 𝑔₂∈𝔽₃^5×11 of the cyclic code with generator polynomial (𝑥−1)ℎ. We compute the action of 𝜎 (the multiplication with 𝑥) and represent this as left multiplication with 𝑧₁₁∈𝔽₃^5×5 on the basis 𝑔₁. If 𝑎=∑⁴_𝑖=0𝑎_𝑖𝑧^𝑖₁₁∈𝔽₃⁵ with 𝑎_𝑖∈𝔽₃, then the entry 𝑎 in 𝐺1 is replaced by ∑⁴_𝑖=0𝑎_𝑖𝑧^𝑖₁₁𝑔₁∈𝔽₃^5×11and analogously for 𝐺2, where we use of course the matrix 𝑔₂ instead of 𝑔₁. Replacing the code by an equivalent one we may choose 𝑎, 𝑏, 𝑐 as orbit representatives of the action of ⟨−𝑧₁₁⟩ on 𝔽^∗₃₅.
A generator matrix of 𝐶 is then given by𝐿0𝑅0𝐺10𝐺20. (3.8) All codes obtained this way are equivalent to the Pless code 𝑃₄₈.

Lemma 3.4. If 𝑝=23, then 𝐶≅𝑃₄₈ or 𝐶≅𝑄₄₈.

Proof. Let 𝜎∈Mon(𝐶) be of order 23. Since (𝑥²³−1)=(𝑥−1)𝑔ℎ∈𝔽₃[𝑥] for irreducible polynomials 𝑔,ℎ of degree 11, 𝔽₃⟨𝜎⟩≅𝔽₃⊕𝔽₃¹¹⊕𝔽₃¹¹. (3.9) Let 𝑒₁,𝑒₂,𝑒₃∈𝔽₃⟨𝜎⟩ denote the primitive idempotents. Then 𝐶=𝐶𝑒₁⊕𝐶𝑒₂⊕𝐶𝑒₃ with 𝐶(𝜎)=𝐶𝑒₁=𝐶𝑒^⟂₁ of dimension 2 and 𝐶𝑒₂=𝐶𝑒^⟂₃≤(𝔽₃¹¹⊕𝔽₃¹¹)². Since all weights of 𝐶 are multiples of 3, this leaves just one possibility for 𝐶(𝜎) (up to equivalence): 𝐶(𝜎)=1²³,0²³,1,0,0²³,1²³,0,1. (3.10) The codes 𝐶𝑒₂ and 𝐶𝑒₃ are codes of length 2 over 𝔽₃¹¹ such that dim(𝐶𝑒₂)+dim(𝐶𝑒₃)=2. Note that the alphabet 𝔽₃¹¹ is identified with the cyclic code of length 23 with generator polynomial (𝑥−1)𝑔 (resp., (𝑥−1)ℎ). These codes have minimum distance 9<15, so dim(𝐶𝑒₂)=dim(𝐶𝑒₃)=1 and both codes have a generator matrix of the form (1,𝑡) (resp., (1,−𝑡⁻¹)) for 𝑡∈𝔽^∗₃₁₁. Going through all possibilities for 𝑡 (up to the action of the subgroup of 𝔽^∗₃₁₁ of order 23) the only codes 𝐶 for which 𝐶(𝜎)⊕𝐶𝑒₂⊕𝐶𝑒₃ have minimum distance ≥15 are the two known extremal codes 𝑃₄₈ and 𝑄₄₈.

Lemma 3.5. If 𝑝=47, then 𝐶≅𝑄₄₈.

Proof. The subcode 𝐶₀∶={𝑐∈𝔽₃⁴⁷|(𝑐,0)∈𝐶} is a cyclic code of length 47, dimension 23, and minimum distance ≥15. Since 𝑥⁴⁷−1=(𝑥−1)𝑔ℎ∈𝔽₃[𝑥] for irreducible polynomials 𝑔,ℎ of degree 23, 𝐶₀ is the cyclic code with generator polynomial (𝑥−1)𝑔 (or equivalently (𝑥−1)ℎ) and 𝐶=⟨(𝐶₀,0),𝟏⟩≤𝔽₃⁴⁸ is the extended quadratic residue code.

3.2. Automorphisms of Order 2

As above let 𝐶=𝐶^⟂≤𝔽₃⁴⁸ be an extremal self-dual ternary code. Assume that 𝜎∈Mon(𝐶) such that the permutational part 𝜋(𝜎) has order 2. Then 𝜎²=±1 because of Lemma 2.1. If 𝜎²=−1, then 𝜎 is conjugate to a block diagonal matrix with all blocks ⁰¹₋₁₀=∶𝐽 and 𝐶 is a Hermitian self-dual code of length 24 over 𝔽₉. Such automorphisms 𝜎 with 𝜎²=−1 occur for both known extremal codes.

If 𝜎²=1, then 𝜎 is conjugate to a block diagonal matrix𝜎∼diag0110^𝑡,1^𝑓,(−1)^𝑎 (3.11) for 𝑡,𝑎,𝑓∈ℕ₀, 2𝑡+𝑎+𝑓=48.

Proposition 3.6. Assume that 𝜎∈Mon(𝐶), 𝜎²=1 and 𝜋(𝜎)≠1. Then either (𝑡,𝑎,𝑓)=(24,0,0) or (𝑡,𝑎,𝑓)=(22,2,2). Automorphisms of both kinds are contained in Aut(𝑃₄₈).

Proof. (1) Wlog 𝑓≤𝑎: Replacing 𝜎 by −𝜎 we may assume without loss of generality that 𝑓≤𝑎.
(2) 𝑓−𝑡∈4ℤ: By Lemma 2.3 the code 𝐶(𝜎)^∗≤𝔽₃^𝑡+𝑓 is a self-dual code with respect to the inner product (𝑥,𝑦)=−∑^𝑡_𝑖=1𝑥_𝑖𝑦_𝑖+∑^𝑓_𝑗=1𝑥_𝑗𝑦_𝑗. This space only contains a self-dual code if 𝑓−𝑡 is a multiple of 4.
(3) 𝑡+𝑓∈{22,24}: The code 𝐶(𝜎)^∗≤𝔽₃^𝑡+𝑓 has dimension (𝑡+𝑓)/2 and minimum distance ≥15/2 and hence minimum distance ≥8. By [8] this implies that 𝑡+𝑓≥22. Since 𝑡+𝑎≥𝑡+𝑓 and (𝑡+𝑎)+(𝑡+𝑓)=48 this only leaves these two possibilities.
(4) 𝑡+𝑓≠22: We first treat the case 𝑓≤14. Then 𝐾^∗≅ker(𝜋_𝑡) is a code of length 𝑓≤14 and minimum distance ≥15 and hence trivial. So 𝜋_𝑡 is injective and 𝐶(𝜎)≅𝐷∶=𝜋_𝑡(𝐶(𝜎))≤𝔽^𝑡₃,dim(𝐷)=11,𝑑(𝐷)≥15−𝑓2. (3.12) Using [8] and the fact that 𝑓−𝑡 is a multiple of 4, this only leaves the cases (𝑡,𝑓)∈{(19,3),(21,1)}. To rule out these two cases we use the fact that 𝐷 is the dual of the self-orthogonal ternary code 𝐷^⟂=𝜋_𝑡(ker(𝜋_𝑓)). The bounds in [9] give 𝑑(𝐷)≤5<(15−3)/2 for 𝑡=19 and 𝑑(𝐷)≤6<(15−1)/2 for 𝑡=21.
If 𝑓≥15, then 𝑡≤7 and 𝐾^∗≅ker(𝜋_𝑡) has dimension 𝑓−𝑡>0 and minimum distance ≥15. This is easily ruled out by the known bounds (see [8]).
(5) If 𝑡+𝑓=24 then either (𝑡,𝑓)=(24,0) or (𝑡,𝑓)=(22,2). Again the case 𝑓>𝑡 is easily ruled out using dimension and minimum distance of 𝐾^∗ as before.
So assume that 𝑓≤𝑡, and let 𝐷=𝜋_𝑡(𝐶(𝜎)) as before. Then dim(𝐷)=12 and using [8] one gets that 𝑡,𝑓∈{(24,0),(22,2),(20,4)}. (3.13) Assume that 𝑡=20. Then there is some self-dual code Λ=Λ^⟂≤𝔽₃²⁰ such that 𝐷^⟂=𝜋_𝑡ker𝜋_𝑓≤Λ=Λ^⟂≤𝐷. (3.14) Clearly also 𝑑(Λ)≥𝑑(𝐷)≥6, so Λ is an extremal ternary code of length 20. There are 6 such codes, and none of them has a proper overcode with minimum distance 6.

Remark 3.7. If 𝜎∈Mon(𝐶) is some automorphism of order 4, then 𝜎²=−1 or 𝜎² has type (24,0,0) in the notation of Proposition 3.6.

Proof. Assume that 𝜎∈Mon(𝐶) has order 4 but 𝜎²≠−1. Then 𝜏=𝜎² is one of the automorphisms from Proposition 3.6 and so 𝜎 is conjugate to some block diagonal matrix 𝜎∼diag0100001000011000^𝑡/2,0110^𝑓/2,0−110^𝑎/2. (3.15) If 𝑡=22 and 𝑓=2 then the fixed code of 𝜎 is a self-dual code in ⟨(1,1,1,1)⟩^𝑡/20𝑥029𝑏9⟨(1,1)⟩^𝑓/2 and 𝐶(𝜎)^∗≤𝔽₃^{𝑡/2+𝑓/2} is a self-dual code with respect to the form (𝑥,𝑦)∶=∑^𝑡/2_𝑖=1𝑥_𝑖𝑦_𝑖−∑^{𝑡/2+𝑓/2}_{𝑖=𝑡/2+1}𝑥_𝑖𝑦_𝑖 which implies that 𝑡/2−𝑓/2 is a multiple of 4, a contradiction.

For the two known extremal codes all automorphisms 𝜎 of order 4 satisfy 𝜎²=−1. It would be nice to have some argument to exclude the other possibility.