Z2Z4-linear codes: generator matrices and duality

• Under the Gray map, quaternary linear codes yield binary Z4-linear codes, including Kerdock and Preparata-like codes whose Z4-duality relates the two families.The Kerdock code’s Z4-dual is the Preparata-like code.Source: §1 · p. 2
• Z2Z4-additive codes are subgroups of Z2^α × Z4^β, a special case of additive codes in a binary Hamming scheme.The coordinate lengths satisfy α + 2β = n.Source: §1 · p. 2
• Z2Z4-linear codes include binary 1-perfect codes, while Kerdock-like and Preparata-like codes cannot have a Z2Z4-linear structure with a nonempty binary part.Source: §1 · p. 2
• Mixed group codes and translation-invariant propelinear codes are broader algebraic settings that also include Z2Z4-additive codes.Abelian translation-invariant propelinear codes are exactly the Z2Z4-linear codes.Source: §1 · pp. 2–3
• The paper gives a general study of Z2Z4-additive codes and corresponding Z2Z4-linear codes, covering fundamental parameters, automorphism groups, generator forms, duality, and self-duality conditions.Section 2 covers definitions, fundamental parameters, and automorphism groups; Sections 3–5 address generator and parity-check forms and duality; Section 6 gives self-duality conditions.Source: §1 · pp. 3–4

• A Z2Z4-additive code is a subgroup of Z2^α × Z4^β, with binary and quaternary coordinates in separate coordinate sets.The Gray map sends its codewords to a binary Z2Z4-linear code of length n = α + 2β.Source: §2 · p. 4
• The extended Gray map is an isometry from Lee distance on the additive code to Hamming distance on its binary image.Each Z4 coordinate maps to two binary coordinates, so the binary length is α + 2β.Source: §2 · p. 4
• The code type (α, β; γ, δ; κ) records the coordinate counts and group structure, with κ the dimension of the binary puncturing of the order-two subcode.The code has group structure Z2^γ × Z4^δ; its binary and quaternary puncturings are linear codes.Source: §2 · p. 4
• Monomially equivalent additive codes have isomorphic binary images, but isomorphic binary images do not necessarily imply monomial equivalence.Monomial equivalence permits coordinate permutations and sign changes in Z4 coordinates.Source: §2 · p. 5

• The proof uses a unique expression of codewords and transforms an associated quaternary linear code to obtain the canonical form up to permutation.The resulting additive code is permutation equivalent to the code generated by the canonical matrix.Source: §3 · pp. 5, 7
• The construction generalizes the known canonical generator-matrix form for quaternary linear codes to Z2Z4-additive codes.The approach views the binary coordinates as quaternary coordinates through the inclusion χ(0)=0, χ(1)=2.Source: §3 · p. 6
• Every Z2Z4-additive code is permutation equivalent to one with a canonical generator matrix whose blocks use binary and quaternary matrices.The theorem identifies the matrix blocks as matrices over Z2 and Z4.Source: §3 · p. 6

• The inner product on a finite abelian group depends on the chosen isomorphism with its dual group, so the resulting pairing is not canonical.For Z2Z4-additive codes, the standard inner product uses the usual Z4 product on quaternary coordinates and twice the usual product on binary coordinates.Source: §4 · pp. 9, 11
• The additive dual of a Z2Z4-additive code is itself additive, and its weight enumerator is related to the original by the MacWilliams identity.The binary images of the code and additive dual are not dual in the binary linear sense.Source: §4 · pp. 11–12
• Changing binary-coordinate ones to twos can change orthogonality, so the additive dual need not match the quaternary dual of the corresponding code.The passage gives an example with α = β = 1 whose vectors are orthogonal under the additive inner product but not under the quaternary one.Source: §4 · p. 12

5 Parity-check matrices of Z2Z4-additive codes

• A homomorphism representing the code can be represented by a matrix that serves as its parity-check matrix or as a generator matrix for its additive dual.Conversely, the homomorphism for the dual gives a matrix that serves as the dual’s parity-check matrix or the original code’s generator matrix.Source: §5 · p. 13
• The additive dual of a code of type (α, β; γ, δ; κ) has type (α, β; α + γ − 2κ, β − γ − δ + κ; α − κ).The parameters are ¯γ = α + γ − 2κ, ¯δ = β − γ − δ + κ, and ¯κ = α − κ.Source: §5 · p. 14
• Two methods construct the additive dual, equivalently a parity-check matrix, from a generator matrix; either method also applies to noncanonical generator matrices.Theorem 3 gives the canonical-matrix construction, while the two general methods are described separately.Source: §5 · pp. 12, 15
• Theorem 3 constructs a generator matrix for the additive dual from a canonical generator matrix, thereby giving a parity-check matrix for the original code.The construction uses matrices over Z2 and Z4 and derives the additive-dual generator from a quaternary dual generator.Source: §5 · pp. 15–16

6 Additive self-dual codes

• An additive self-dual code has type (2κ, β; β + κ − 2δ, δ; κ), with |C| = 2^(κ+β) and |Cb| = 2^(κ+β−δ).Here Cb is the subcode containing all order-two codewords.Source: §6 · p. 18
• For codewords (x | y) in an additive self-dual code, even w(x) implies p(y) ≡ 0 (mod 4), odd w(x) implies p(y) ≡ 2 (mod 4), and (0 | 2) is a codeword.p(y) counts the odd, or order-four, coordinates of y.Source: §6 · p. 18
• The projection (Cb)X of the order-two subcode is a binary self-dual code.It has dimension κ and length 2κ.Source: §6 · pp. 18–19
• For some 0 ≤ r ≤ κ, each word in CY occurs 2^r times in C, and |CY| ≥ 2^β.The proof sets r to the dimension of the binary projection of the subcode whose quaternary coordinates are zero.Source: §6 · p. 19
• Additive self-duality holds for the stated canonical family when δ ≤ κ and β − 2δ ≡ 0 (mod 4) (Lemma 10), and when δ < κ and β − 2δ ≡ 2 (mod 4) (Lemma 12).For Lemma 12, C = Φ(C) is antipodal and CX is not self-dual.Source: §6 · pp. 20, 24
• A self Z2Z4-dual code is antipodal exactly when its binary-coordinate projection has only even weights; its Gray image then also has only even weights.For any word, Gray-image weight parity matches the parity of its binary-coordinate weight.Source: §6 · pp. 20–21
• For additive self-dual codes, binary self-duality of CX is equivalent to quaternary self-duality of CY; when both hold, C is the direct product CX × CY.Conversely, the direct product of binary and quaternary self-dual codes is additive self-dual.Source: §6 · pp. 21–22
• The minimum-cardinality, minimum-coordinate additive self-dual code whose Gray image is antipodal but whose CX is not self-dual has type (4, 4; 4, 1; 2).Such codes require α ≥ 4 and β ≥ 4.Source: §6 · p. 23