Propagation Rules of Subsystem Codes

We demonstrate propagation rules of subsystem code constructions by extending, shortening and combining given subsystem codes. Given an [[n, k, r, d]]q subsystem code, we drive new subsystem codes with parameters [[n + 1, k, r,≥ d]]q, [[n − 1, k + 1, r,≥ d − 1]]q , [[n, k − 1, r + 1, d]]q . We present the short subsystem codes. The interested readers shall consult our companion papers for upper and lower bounds on subsystem codes parameters, and introduction, trading dimensions, families, and references on subsystem codes [1], [2], [3] and references therein. Subsystem Codes. Let H be the Hilbert space C n = C ⊗ C q ⊗ ... ⊗ C . Let Q be a quantum code such that H = Q⊕Q, where Q is the orthogonal complement of Q. Recall definition of the error model acting in qubits [4], [3]. We can define the subsystem code Q as follows Definition 1: An [[n, k, r, d]]q subsystem code is a decomposition of the subspace Q into a tensor product of two vector spaces A and B such that Q = A⊗B, where dimA = k and dimB = r. The code Q is able to detect all errors of weight less than d on subsystem A. Subsystem codes can be constructed from the classical codes over Fq and Fq2 . The Euclidean construction of subsystem code is given as follows [1], [3]. Lemma 2 (Euclidean Construction): If C is a kdimensional Fq-linear code of length n that has a kdimensional subcode D = C ∩ C and k + k < n, then there exists an [[n, n− (k + k), k − k,wt(D \ C)]]q subsystem code. I. SUBSYSTEM CODES VERS. CO-SUBSYSTEM CODES In this section we show how one can trade the dimensions of subsystem and co-subsystem to obtain new codes from a given subsystem or stabilizer code. The results are obtained by exploiting the symplectic geometry of the space. A remarkable consequence is that nearly any stabilizer code yields a series of subsystem codes. Our first result shows that one can decrease the dimension of the subsystem and increase at the same time the dimension of the co-subsystem while keeping or increasing the minimum distance of the subsystem code. Theorem 3: Let q be a power of a prime p. If there exists an ((n,K,R, d))q subsystem code with K > p that is pure to d, then there exists an ((n,K/p, pR,≥ d))q subsystem code that is pure to min{d, d}. If a pure ((n, p,R, d))q subsystem code exists, then there exists a ((n, 1, pR, d))q subsystem code. Proof: See [1], [2] Replacing Fp-bases by Fq-bases in the proof of the previous theorem yields the following variation of the previous theorem for Fq-linear subsystem codes. Theorem 4: Let q be a power of a prime p. If there exists a pure Fq-linear [[n, k, r, d]]q subsystem code with r > 0, then there exists a pure Fq-linear [[n, k + 1, r − 1, d]]q subsystem code. Proof: See [1], [2] Theorem 5 (Generic methos): If there exists an (Fq-linear) [[n, k, d]]q stabilizer code that is pure to d, then there exists for all r in the range 0 ≤ r < k an (Fq-linear) [[n, k− r, r,≥ d]]q subsystem code that is pure to min{d, d} . If a pure (Fqlinear) [[n, k, r, d]]q subsystem code exists, then a pure (Fqlinear) [[n, k + r, d]]q stabilizer code exists. Proof: See [1], [2] Using this theorem we can derive many families of subsystem codes derived from families of stabilizer codes as shown in Table 1 II. PROPAGATION RULES Let C1 ≤ Fq and C2F n q be two classical codes defined over Fq . The direct sum of C1 and C2 is a code C ≤ F q defined as follows C = C1 ⊕ C2 = {uv | u ∈ C1, v ∈ C2}. (1) In a matrix form the code C can be described as C = ( C1 0 0 C2 ) An [n, k1, d1]q classical code C1 is a subcode in an [c, k2, d2]q if every codeword v in C1 is also a codeword in C2, hence k1 ≤ k2. We say that an [[n, k1, r1, d1]]q subsystem code Q1 is a subcode in an [[n, k2, r2, d2]]q subsystem code Q2 if every codeword |v〉 in Q1 is also a codeword in Q2 and k1 + r1 ≤ k2 + r1. Notation. Let q be a power of a prime integer p. We denote by Fq the finite field with q elements. We use the notation (x|y) = (x1, . . . , xn|y1, . . . , yn) to denote the concatenation of two vectors x and y in Fq . The symplectic weight of (x|y) ∈ F q is defined as swt(x|y) = {(xi, yi) 6= (0, 0) | 1 ≤ i ≤ n}. 2 Family Stabilizer [[n, k, d]]q Subsystem [[n, k − r, r, d]]q , k > r ≥ 0 Short MDS [[n, n− 2d + 2, d]]q [[n, n− 2d + 2 − r, r, d]]q Hermitian Hamming [[n, n− 2m, 3]]q m ≥ 2, [[n, n− 2m− r, r, 3]]q Euclidean Hamming [[n, n− 2m, 3]]q [[n, n− 2m− r, r, 3]]q Melas [[n, n− 2m,≥ 3]]q [[n, n− 2m− r, r,≥ 3]]q Euclidean BCH [[n, n− 2m⌈(δ − 1)(1 − 1/q)⌉,≥ δ]]q [[n, n− 2m⌈(δ − 1)(1 − 1/q)⌉ − r, r,≥ δ]]q Hermitian BCH [[n, n− 2m⌈(δ − 1)(1 − 1/q)⌉,≥ δ]]q [[n, n− 2m⌈(δ − 1)(1 − 1/q)⌉ − r, r,≥ δ]]q Punctured MDS [[q − qα, q − qα− 2ν − 2, ν + 2]]q [[q − qα, q − qα− 2ν − 2 − r, r, ν + 2]]q Euclidean MDS [[n, n− 2d + 2]]q [[n, n− 2d + 2 − r, r]]q Hermitian MDS [[q − s, q − s− 2d + 2, d]]q [[q − s, q − s− 2d + 2 − r, r, d]]q Twisted [[q, q − r − 2, 3]]q [[q, q − r − 2 − r, r, 3]]q Extended twisted [[q + 1, q − 3, 3]]q [[q + 1, q − 3 − r, r, 3]]q Perfect [[n, n− s− 2, 3]]q [[n, n− s− 2 − r, r, 3]]q [[n, n− s− 2, 3]]q [[n, n− s− 2 − r, r, 3]]q Fig. 1. Families of subsystem codes from stabilizer codes We define swt(X) = min{swt(x) |x ∈ X,x 6= 0} for any nonempty subset X 6= {0} of F q . The trace-symplectic product of two vectors u = (a|b) and v = (a|b) in F q is defined as 〈u|v〉s = trq/p(a ′ · b− a · b), where x · y denotes the dot product and trq/p denotes the trace from Fq to the subfield Fp. The trace-symplectic dual of a code C ⊆ F q is defined as Cs = {v ∈ F q | 〈v|w〉s = 0 for all w ∈ C}. We define the Euclidean inner product 〈x|y〉 = ∑n i=1 xiyi and the Euclidean dual of C ⊆ Fq as C = {x ∈ Fq | 〈x|y〉 = 0 for all y ∈ C}. We also define the Hermitian inner product for vectors x, y in Fq2 as 〈x|y〉h = ∑n i=1 x q i yi and the Hermitian dual of C ⊆ Fq2 as Ch = {x ∈ Fq2 | 〈x|y〉h = 0 for all y ∈ C}. Theorem 6: Let C be a classical additive subcode of F q such that C 6= {0} and let D denote its subcode D = C∩Cs . If x = |C| and y = |D|, then there exists a subsystem code Q = A⊗B such that i) dimA = q/(xy), ii) dimB = (x/y). The minimum distance of subsystem A is given by (a) d = swt((C +Cs)−C) = swt(Ds −C) if Ds 6= C; (b) d = swt(Ds) if Ds = C. Thus, the subsystem A can detect all errors in E of weight less than d, and can correct all errors in E of weight ≤ ⌊(d−1)/2⌋. A. Extending Subsystem Codes We derive new subsystem codes from known ones by extending and shortening the length of the code. Theorem 7: If there exists an ((n,K,R, d))q Clifford subsystem code with K > 1, then there exists an ((n+1,K,R,≥ d))q subsystem code that is pure to 1. Proof: We first note that for any additive subcode X ≤ F 2n q , we can define an additive code X ′ ≤ F q by X ′ = {(aα|b0) | (a|b) ∈ X,α ∈ Fq}. We have |X | = q|X |. Furthermore, if (c|e) ∈ Xs , then (cα|e0) is contained in (X )s for all α in Fq, whence (Xs) ⊆ (X )s . By comparing cardinalities we find that equality must hold; in other words, we have (Xs) = (X )s . By Theorem 6, there are two additive codes C and D associated with an ((n,K,R, d))q Clifford subsystem code such that |C| = qR/K and |D| = |C ∩Cs | = q/(KR). We can derive from the code C two new additive codes of length 2n+2 over Fq , namely C and D = C∩(C)s . The codes C and D determine a ((n + 1,K , R, d))q Clifford subsystem code. Since D = C ∩ (C)s = C ∩ (Cs) = (C ∩ Cs), we have |D| = q|D|. Furthermore, we have |C| = q|C|. It follows from Theorem 6 that (i) K ′ = q/ √ |C′||D′| = q/ √ |C||D| = K , (ii) R = (|C|/|D|) = (|C|/|D|) = R, (iii) d = swt((D)s \ C) ≥ swt((Ds \ C)) = d. Since C contains a vector (0α|00) of weight 1, the resulting subsystem code is pure to 1. Corollary 8: If there exists an [[n, k, r, d]]q subsystem code with k > 0 and 0 ≤ r < k, then there exists an [[n+1, k, r,≥ d]]q subsystem code that is pure to 1.