Logic Functions and Quantum Error Correcting Codes

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by  Yajie Xu

Logic Functions and Quantum Error Correcting
Codes
Yajie Xu1, Zhi Ma1, Chunyuan Zhang1, Xin Lu¨2
1
Department of Information Research, Information Engineering University,
8
Zhengzhou, 450002, China
0
e-mail:[email protected]
0
2
2 Informatization Institute, StateInformation Center, Beijing, 100045, China.

n
a
J Abstract. In this paper, based on the relationship between logic func-
6
tions and quantum error correcting codes(QECCs), we unify the con-

struction of QECCs via graphs, projectors and logic functions. A con-
]
h structionofQECCsoveraprimefieldFp isgiven,andoneoftheresults
p given by Ref[8] can be viewed as a corollary of one theorem in this pa-
- per. With the help of Boolean functions, we give a clear proof of the
t
n existence of a graphical QECC in mathematical view, and find that the
a existence of an [[n,k,d]] QECC over Fp requires similar conditions with
u that depicted in Ref[9]. The result that under the correspondence de-
q finedinRef[17],every[[n,0,d]]QECCoverF2 correspondingtoasimple
[
undirected graph has a Boolean basis state, which is closely related to

4 the adjacency matrix of thegraph, is given.
v After a modification of the definition of operators, we find that some
5 QECCsconstructedviaprojectors depictedinRef[11]canhaveBoolean
0 basis states. A necessary condition for aBoolean function being used in
6 the construction via projectors is given. We also give some examples to
3
illustrate our results.
.
2
1
7 1 Introduction
0
:
v Quantum error correcting codes(QECCs)have have received more and more at-
i tentionfornearestfewdecadessincethetheoryofquantumerrorcorrectionwas
X
put forward[1,2,3,4]. One of the central tasks in the theory of QECCs is how to
r
a constructthem,andthefirstsystematicmathematicalconstructionisgivenin[4]
in the binary case and then generalized in [5,6]. Many good non-binary QECCs
havebeen constructedbyusingclassicalerror-correctingcodesoverF orF (q
q q2
is a power of an odd prime ) with special orthogonal properties. Besides these,
many quantum codes are constructed via tools including graphs and Boolean
functions[7,8,9,10]. In this paper, we unify the two tools.
In Ref[7], the author constructed [[n,0,d]] QECCs using Boolean functions
with n variablesandaperiodic propagationcriterion(APC)distance d,andgave
an algorithm to compute the APC distance of a Boolean function, orbits of
Boolean functions of the same APC distance are also studied. In this paper, a
construction of quantum codes of dimension more than 0 is given basing on the
relationship between logic functions and quantum codes. In Ref[8], the authors
constructedquantumcodesbygivingK basisstatesbasedonagraphstateasso-
ciatedwithagraph,andeachbasisstatecorrespondstoasubsetofthevertexes,
which can be viewed as a corollary of this paper with weaker requirements.
In Ref[9], the authors initiated the construction of QECCs via the the con-
struction of matrixes with some properties and proved the necessary and suffi-
cient conditions for the graph that the resulting code corrects a certain number
oferrorsinphysicsview.InRef[10],theauthorgaveanotherproofinmathemat-
ical view. Based on the mathematical proof and the close relationship between
logic functions and matrixes, properties that a logic function should have in or-
der to construct a QECC is systematically studied.and find that the existence
of an [[n,k,d]] QECC over F requires similar conditions with that depicted in
p
Ref[9]. Under the correspondence defined in Ref[17], we find that every [[n,0,d]]
over F corresponding to a simple undirected graph with the adjacency matrix
2
Γn×n hasthebasisstate|ψfi=2−n2 (−1)f(x)|xi,wheref(x)= 12xΓxT.We
x∈PF2n
also give an example to illustrate our result.
In Ref[11], the author describes a common mathematical framework for the
design of QECCs basing on the correspondence between Boolean functions and
projection operators. We point out that in some conditions, the basis states of
the QECCs under the framework can have the probability vector of the form
2−n2(−1)f(x), where f(x) is a Boolean function.
2 Preliminaries
We oftenexpressanerror(alsocanbe viewedasanoperator)operatingonCqn
asE =iλX Z ,wherea,barevectorsoflengthnoverF ,whereq isapower
(a,b) a b q
of a prime p. And
n
E(a,b)|x1x2···xni=iλζtrq/p(jP=1bjxj)|x′1x′2···x′ni (1)
where x′ =x +a , ζ is a p-th primitive root of 1.
j j j
Especially, an error acting on a n-qubit state in C2n has simpler forms .
Definition 1. Operators E associated with binary vectors (a,b) ∈ F2n are
(a,b) 2
defined by
E =e ⊗...⊗e =ia·bX Z (2)
(a,b) 1 m a b
I , a =0,b =0.
2 i i
where e =σx, ai =1,bi =0..
i σz, ai =0,bi =1.
σ , a =1,b =1.
y i i
Definition2. The weight of an error E = iλX Z is defined by the sym-
(a,b) a b
plectic weight of two vectors a,b of length n, i.e.,
W (a,b)=♯{i|1≤i≤n,(a ,b )6=(0,0)} (3)
s i i
WegeneralizethedefinitionofAPC(aperiodicpropagationcriterion)distance
of a Boolean function in Ref[7] to F .
p
Definition 3. The APC (aperiodic propagation criterion)distance of a logic
function f over F is defined by the smallest nonzero w (a,b), where a,b ∈ Fn
p s p
such that
ζf(x)+f(x−a)+b·x 6=0 (4)
xX∈Fn
p
where ζ is a p-th primitive root of 1.
LetG=(V,E)beagraphwithvertexsetV =X∪Y andedgesetE =V×V.
Each edge uv ∈ E(u,v ∈ V) is assigned a weight a (= a ) ∈ F . Therefore,
uv vu p
such a graph G corresponds to a symmetric matrix over F A =(a )
p n×n uv u,v∈V
.
For two subsets S and T of V, we denote A as the submatrix of A with
S,T
size |S|×|T|
A =(a )
S,T uv u∈S,v∈T
Similarly, a vector in the vector space F|V| = Fn+k is denoted by a column
p p
vector
dv1
 dv2 
dV = . ={dv|v ∈V}
.
 . 
 
dvn+k
 
where dvi ∈Fp. For a subset S of V , we denote dS ={ds|s∈S}∈FpS, and OS
a vector of length |S| with every coordinate equal to 0.
Let E be a subset of Y with d−1 elements, I =Y \E, then
A A A
XX XE XI
A=AEX AEE AEI
A A A
IX IE II
 
Lemma 1. [9] Suppose X,Y are two disjoint sets. |X| = k,|Y| = n ≥ k +
2d−2,d ≥ 2, A = (a ) is a symmetric matrix with vanishing diagonal
ij i,j∈X∪Y
entries. For arbitrary E ⊆Y,|E|=d−1, if
A dX +A dE =OI (5)
IX IE
with I =Y \E implies that
dX =OX and A dE =OX (6)
XE
Then there exists an [[n,k,d]] quantum code.
Definition 4. [11] We define the Zset of a Boolean function f by
f
Zset ={a| f(x)f(x+a)=0}
f
x∈PF2n
Lemma 2. [11] If the weight of the Boolean function f with n variables is M,
and M ≤2n−1, then Zsetf ={a|r (a)=2n−4M}, where r (a) is the autocor-
f f
rection function of f(v) at a, i.e., r (a)= (−1)f(x)+f(x+a).
f
x∈PF2n
Lemma 3. [11] An ((n,M,2))-QECC is determined by a Boolean function f
with the following properties
1) f is a function of n variables and has weight M.
2) The Zset contains the set {α ,α ,...,α ,α +α ,...,α +α }) and
f 1 2 2n 1 n+1 n 2n
the matrix A = (α ,α ,...,α ) has the property that any two rows
f 1 2 2n n×2n
have symplectic inner product zero and all the rows are independent.
In the above lemma, the QECC is constructed by giving the projector P =
f(P ,P ,···,P ) onto the code, and P is constructed in the sense of a logic of
1 2 n
projection operators given in Ref[11], where {P = 1(I +E )|1 ≤ i ≤ n},
n+1−i 2 γi
γ is the i − th row of the matrix A . From the symplectic orthogonality of
i f
the rows of A , we have {P |1 ≤ i ≤ n} are pairwise commutative, and the
f i
error-correcting ability of the QECC is ensured by the properties of Zset . An
f
arbitrary error e acting nontrivially on one qubit only takes the projector P
f(x)
to P , i.e., eP e = P , where t is an element in Zsetf, and P
f(x+t) f(x) f(x+t) f(x+t)
is orthogonalto P .
f(x)
3 logic functions and quantum states
For a logic function f(x) with n variables over F , it corresponds to a vector
p
s=p−n2ζf(x), which can be interpreted as the probability distribution vector of
the quantum state
|ψfi=p−n2 ζf(x)|xi (7)
xX∈Fn
p
Specially,ifastatehastheformof2−n2 (−1)f(x)|xi,wheref(x)isaBoolean
x∈PF2n
function, we call it a Boolean state corresponding with f(x).
ThenifanerrorE actsonthestate|ψ i,ittakesittoanotherstatewhich
(a,b) f
isproportionaltop−n2 ζf(x−a)+b·x|xi, whichcanalsobe expressedinterms
x∈PFpn
of a logic function, f(x)→f(x−a)+b·x.
Let K Boolean functions be g (x) = f(x)+β ·x, and for 1 ≤ i < j ≤ K,
i i
βi 6= βj , define K quantum states as |ψii = p−n2 ζgi(x)|xi, then we have
x∈PFpn
the following theorem.
Theorem 1. The subspace spanned by {|ψ i|1 ≤ i ≤ K} is an ((n,K,d′))
i p
quantumcode,whered′ =min{W (u,v)|thereexist1≤i≤j ≤K suchthatW (u,v+
s s
β +β )≥d}, where d is the APC distance of f(x).
i j
Proof. We only need to prove that for any error ε acting nontrivially on
d
less than d qubits, hψ |ε |ψ i = f(ε )δ for all 1 ≤ i,j ≤ K. Without lose
i d j d ij
of generosity we assume that ε = X Z for some pair of vectors in Fn with
d u v p
W (u,v)<d′.
s
We have
hψ |ε |ψ i∝ ζf(x−u)+(βi+βj+v)·x+f(x). (8)
i d j
xX∈Fn
p
Since W (u,v+β +β )<d, so hψ |ε |ψ i=0.
s i j i d j
Then we verify that hψ |ε |ψ i only depends on ε .
i d i d
hψ |ε |ψ i∝ ζf(x−u)+v·x+f(x) =0 (9)
i d i
xX∈Fn
p
So, {|ψ i|1≤i≤K} span an ((n,K,d′)) quantum code.♯
i p
4 logic functions and graphical QECCs
4.1 quadratic Boolean functions of the form 1xΓxT
2
Now we consider a classof quadraticlogic function correspondingwith a simple
undirected graph.
If f(x) is a quadratic Boolean function and can be represented as f(x) =
1xΓxT,where Γ is a symmetric matrix with elements in F and vanishing
2 n×n 2
diagonal entries, then the state |ψ i can be viewed as a graph state because Γ
f
can be viewed as the adjacency matrix of a graph G = (V,E), where V and E
denote the set of vertices and edges respectively and |V| = n. If we label every
vertex of the graph G of n vertexes from 1 to n, then every vertex corresponds
withonequbit,andtheerrorε =X Z canbewrittenasX Z ,whereω,δ are
d u v ω δ
subsets of V.
Consider the operator G = X Z , where N represents the neighbor-
a a b a
b∈QNa
hoodofthevertexaandisdenotedbyN ={v ∈V|Γ =1},anditwasshown
a av
in Ref[12] that G |ψ i=|ψ i, so X Z |ψ i=|ψ i.
a f f ω Nω f f
Choose K subsets of V {C |1 ≤ i ≤ K}, we define K pair-wise orthogonal
i
quantum states[8] as
|ψii=2−n2 (−1)gi(x)|xi=ZCi|ψi. (10)
xX∈Fn
2
whereandcorrespondstoavectorβ oflengthn,thereforeg (x)canbeexpressed
i i
as g (x) = f(x)+β ·x. It was shown that [8] G |ψ i = −|ψ i if a ∈ C and
i i a i i i
G |ψ i=|ψ i if a∈C otherwise.
a i i i
Definition 5. [8] The d−uncoverable set D that contains all the subsets of V
d
which can’t be covered by less than d vertices is denoted by
D =2V −{δ△N ||ω δ|<d}
d ω
S
where ω△δ denotes the symmetric difference of two sets ω,δ, i.e., ω△δ = ω∪
δ−ω∩δ, and N denotes the neighborhood of the ω, i.e., for every element v
ω
in N , there exist an element v′ in ω such that Γ =1.
ω vv′
Corollary 1. If C ={C ,C ,...,C } satisfies the following two conditions,
1 2 K
(1) Ø∈C; (2) C △C ∈D .
i j d
thenthesubspacespannedbythebasis{|ψ i=Z |ψi|1≤i≤K}isan((n,K,d))
i Ci
code,
Proof.We chooseaBooleanfunctionf(x)= 1xΓxT with APCdistance d,and
2
fromTheorem1,{|ψ i|1≤i≤K}spana((n,K,d)),whered=min{W (u,v)|thereexist1≤
i s
i≤j ≤K such that W (u,v+β +β )≥d }.
s i j
For an correctable error ε =X Z =X Z , W (u,v)≤d−1, we have
d u v ω δ s
hψ |ε |ψ i ∝hψ|Z Z |ψi (11)
j d i δ△Nω Ci△Cj
∝ (−1)f(x)+f(x+u)+(v+βi+βj)·x (12)
xX∈Fn
2
And W (u,v+β +β )<d, so
s i j
(−1)f(x)+f(x+u)+(v+βi+βj)·x =0.
x∈PF2n
Then we have δ△N 6=C △C , in other words, C △C ∈D .♯
ω i j i j d
It should be noted here that in Ref[8] ,the authors gave three conditions for
the existence of an ((n,K,d)) over F quantum code, and we consider them
2
unnecessary.
4.2 quadratic logic functions of the form 1(c,x)A(c,x)T
2
Consider an (n+k)×(n+k) symmetric matrix A with elements in F and
p
vanishing diagonal entries, then for every vector c of length k with elements
in F , f(c,x) = 1(c,x)A(c,x)T is a logic function of n variables, where x =
p 2
(x ,x ,···,x ) is a vector of n variables. Notice that the degree of f(x) is at
1 2 n
most two.
Now, we consider the sufficient conditions for the set of states {|ψ i =
i
p−n2 ζf(ci,x)|xi|ci ∈ Fpk} that can span an [[n,k,d]]p quantum code, i.e.,
x∈PFpn
the required properties of the Boolean function. Basing on Lemma 1,we have
the following theorem.
Theorem 2. Suppose C,X are two disjoint sets. |C| = k,|X| = n,d ≥ 2,
A = (a ) is a symmetric matrix with elements in F and vanishing
ij i,j∈C∪X p
diagonal entries. For arbitrary E ⊆ X,|E| = d−1, I = X \E, if the rows of
A are linear independent , and
EI
A dC +A dE =OI (13)
IC IE
implies that
dC =OC (14)
. Then the subspace spanned by {|ψ i} is an [[n,k,d]] code over F , where ζ is
i p p
a p−th primitive root of 1.
Proof.WefirstprovethatifA dC+A dE =OI whereI =X\E implies
IC IE
that dX =OX , then for different i,j, hψ |ψ i=0.
j i
hψ |ψ i∝ ζf(ci,x)−f(cj,x) (15)
j i
xX∈Fn
p
A A
Since A can be expressed as A= CC CX , then
(cid:18)AXC AXX(cid:19)
ζf(ci,x)−f(cj,x) ∝ ζxAXC(ci−cj)T (16)
xX∈Fn xX∈Fn
p p
so hψ |ψ i=6 0 iff A (c −c )T =OX.
j i XC i j
Seeking a contradiction, we suppose A (c −c )T = OX, then there exist
XC i j
I ⊆ X such that A (c −c )T = OI. Let dE = OE, then A (c −c )T +
IC i j IC i j
A dE = OI, which satisfies Eq.(13), so (c −c )T = OC which is impossible
IE i j
becauseofi6=j.Wecometotheresultthat{|ψ i}spanasubspaceofdimension
i
pk.
Thenweprovethesubspacespannedby{|ψ i}isan[[n,k,d]] quantumcode,
i p
i.e., for an error ε =X Z with W (a,b)≤d−1, hψ |ε |ψ i=f(ε )δ .
d a b s j d i d ij
hψ |ε |ψ i∝ ζbx+xAXC(ci−cj)T+aAXXxT (17)
j d i
xX∈Fn
p
Letε actsonqubitscorrespondingwithasubsetE ⊆X,anda ,b arevectors
d E E
of length d−1. For simplicity, we denote variables in I as y, variables in E as
z. Then
hψj|εd|ψii∝ ζbEz+zAEC(ci−cj)T+zAEEaTE+yAIC(ci−cj)T+yAIEaET (18)
xX∈Fn
p
Fordifferenti,j,considerlineartermsofy,ifA (c −c )T+A a T 6=OI,
IC i j IE E
then hψ |ε |ψ i = 0. If A (c −c )T +A a T = OI, which satisfies Eq.(13),
j d i IC i j IE E
so (c −c )T =OC which contradicts the fact that c ,c are different.
i j i j
Then we verify hψ |ε |ψ i only depends on ε .
i d i d
hψi|εd|ψii∝ ζbEz+zAEEaTE+yAIEaET
x∈PFpn
Since the rows of A are independent, then if A a T = 0, we can know
EI IE E
aE =0, thus hψi|εd|ψii∝ ζbEz. Because bE 6=0, hψi|εd|ψii=0.
x∈PFpn
So, {|ψ i|1≤i≤2k} span an [[n,k,d]] quantum code. ♯
i p
4.3 Graphical [[n,0,d]] QECC
Consider the adjacency matrix(n × n) Γ of a graph, then the rows of A =
(ωI|Γ) can span an self-dual additive code C over GF(4)={0,1,ω,ω2}, where
ω2+ω+1=0. And C is equivalent to a graph code D[17].
Let α ,β are the i−th column of I and Γ respectively, in fact, X Z are
i i αi βi
stabilizers of D, if we can find a Boolean function f(x) satisfying the following
equations:
f(x+α )=β x, for 1≤i≤n (19)
i i
then we can state that |ψi=2−n2 (−1)f(x)|xi is the basis state. So we find
x∈PF2n
that the graph code is equivalent to [[n,0,d]], where d is the APC distance of
f(x).
Example 1. Consider a complete graph of 4 vertices, then matrix
ω 0 0 0 0111
0 ω 0 0 1011
A=
0 0 ω 0 1101
 
0 0 0 ω 1110
 
After computation, we find that f(x) = x x +(x +x )(x +x )+x x , the
1 2 1 2 3 4 3 4
APCdistanceoff(x)is2,so{|ψi=2−2 (−1)f(x)|xi}spana[[4,0,2]]QECC.
x∈PF24
Infact,underthecorrespondencedefinedinRef[17],everysimpleundirected
graph with adjacency matrix Γ corresponds to an [[n,0,d]] QECC over F
n×n 2
with the basis state |ψi=2−n2 (−1)f(x)|xi, where f(x)= 1xΓxT.
2
x∈PF2n
5 Boolean functions and projectors
InRef[11],theauthorsconstructedquantumerrorcorrectingcodesviathe tools
of projectors and Boolean functions. They first redefine a logic of projectors,
thenonthe assumptionthattheycanconstructacertainmatrix,whichsatisfies
some propertiescorrespondingto the Z of aBooleanfunction f,finally they
setf
construct a projector onto a quantum code.
Inthissection,werefinetheprojectorsanotherlogicofprojectors,andcome
totheresultthatinsomeconditions,thequantumcodesunderthe construction
which is similar to that givenRef[11] has Booleanbasis states. Now we give our
definition of operator , which is denoted by E′ .
(a,b)
Definition 6. Operators E′ associated with binary vectors (a,b) ∈ F2n are
(a,b) 2
redefine by
E′ =e′ ⊗...⊗e′ (20)
(a,b) 1 n
where e′j =iajbjej for 1≤j ≤n. In other words, E(′a,b) =XaZb.
Basing on the definition of the logic of projection operators in Ref[11], we
define another logic as the following definition.
Definition 7. Let P = X Z ,P′ = X Z ,P′′ = X Z are three projection
a b a′ b′ a′′ b′′
operators,wherea,a′,a′′,b,b′,b′′ arevectorsoflengthn.ThenwedefineP∨P′ =
X Z +X Z , P ∧P′ = (−1)a′bX Z , P˜ = I −P and (P ∨P′)∧P′′ =
a b a′ b′ a+a′ b+b′
(−1)a′′bX Z +(−1)a′′b′X Z .
a+a′′ b+b′′ a′+a′′ b′+b′′
Definition 8. [11] Given an arbitrary Boolean function f(x ,x ,···,x ), we
1 2 n
define the Projection function fˆ(P ,P ,···,P ) in which x is replaced by P ,
1 2 n i i
multiplication, summation and not operation in Boolean logic are replaced by the
meet, join and tilde operation in the projection logic described in Definition 7
respectively.
We denote Pci as P if c =0, and P˜ if c =1.
i i i i i
If we can construct matrix A = (A|B) as in Lemma 3, where A and B are
f
blocks of A of size n×n with the i-th row vectors α ,β , corresponding with a
f i i
Booleanfunctionf(x) with n variables,then we redefinethe operationoperator
P = 1(I+E′ ).Theprojectorfˆ(P ,P ,···,P )isstillaprojectorontoan
n+1−i 2 γi 1 2 n
((n,M,2)) QECC, where M is the Hamming weight of f(x).
The projector P onto a QECC Q has the form P = |ψihψ|, where |ψi
run over all the basis states of Q. Without lose of generoPsity, we assume the
vector (t ,t ,···,t ) is an element of the support of f, in fact, every element
1 2 n
in the support of f corresponds to a basis state. Then the term corresponds
to (t1,t2,···,tn) in P = fˆ(P1,P2,···,Pn) is P1t1,P2t2,···,Pntn, which can be
written as
n
2−n (−1)λ(d,t)|x+ d α ihx| (21)
i i
dX∈FnxX∈Fn Xi=1
2 2
n n
, where λ(d,t)=( d β )x+ t d + d d α β .
i i i i j k j k
iP=1 iP=1 1≤jP<k≤n
Now we consider properties that the Boolean function f should
(t1,t2,···,tn)
have if P1t1,P2t2,···,Pntn can be written as |ψ(t1,t2,···,tn)ihψ(t1,t2,···,tn)|, where
|ψ i is a Boolean state corresponding with f .
(t1,t2,···,tn) (t1,t2,···,tn)
Forsimplicity,wewritef˘inplaceoff ,|ψ˘iinplaceof|ψ i),
(t1,t2,···,tn) (t1,t2,···,tn)
and
|ψ˘ihψ˘|∝ (−1)f˘(x)+f˘(x+s)|x+sihx| (22)
sX∈FnxX∈Fn
2 2
Then we have the following theorem.
Theorem 3. Aisinvertible, f˘(x)isisquadraticandf˘(x)+f˘(x+α )=β x+t .
i i i
Proof. Since s(in Eq.(22)) and d(in Eq.(21)) run over Fn, we require that
2
α ,α ,···,α are linear independent, and
1 2 n
f˘(x)+f˘(x+α )=β x+t . (23)
i i i
n n n
Then for arbitrary d ∈ Fn, f˘(x) + f˘(x + d α ) = ( d β )x+ d t +
2 i i i i i i
iP=1 iP=1 iP=1
d d α β , which coincides with λ(d,t).
j k j k
1≤jP<k≤n
Since the right part of Eq.(23) is an affine Boolean function, we know that
f˘(x) is quadratic. ♯
If a QECC has Boolean states, then the study of the QECC can again be
convertedto the study of Booleanfunctions correspondingwith them, we say it
is possible.
Example 2. Define a Boolean function g(y ,y ,y ,y ) = (y + y + y )(y +
1 2 3 4 1 2 3 1
y +y ), then g(y) is partially bent, and |Supp(g)| = 4, then for every a ∈ F4
2 4 2
with r (a) = 0 is in Zsetg, and Supp(g) = {s = (1000),s = (0100),s =
g 1 2 3
(0011),s = (1111)}. Let a be a unitary vector of length 4 with 1 in the i−th
4 i
coordinate and 0 elsewhere. Since g(y+a ) = g(y)+b x+c (1 ≤ i ≤ 4, c =
i i i
(1100)), we construct the matrix A as
g
10000011
01000011
A =
g 00101101
 
00011110
 
where the i−th row of A is v = (a ,b ). We can easily verify that all the
g i i i
rows of A are independent and any two rows have symplectic product zero
g
because the right four columns of A form a symmetric matrix. Express A as
g g
A =[x ,x ,···,x ],thenforeveryωwithW (ω)≤1,A ∗ωT isinZsetg.After
g 1 2 8 s g
computation, we have f˘ =g+y ,f˘ =g+y ,f˘ =g+y +y +y +y ,f˘ =
1 2 2 1 3 1 2 4 3 4
g +y + y . We can see that for different i,j, f˘ − f˘ are linear terms. And
3 4 i j
{2−2 (−1)f˘i(x)|xi}spansa[[4,2,2]]code,whichmeetsthequantumsingleton
x∈PF24
bound, and therefore is an MDS code.
In fact, for every function f(y) with 2m variables of the form f(y) = (y +
1
y +···+y +y )(y +y +···+y +y ) (which is a partially
2 2m−2 2m−1 1 2 2m−2 2m
bent function[16]), we can find |Supp(f)| Boolean functions f˘ satisfying that
i
{2−m (−1)f˘i(x)|xi} spans a [[2m,2m−2,2]] MDS code.
x∈PF22m
Becausea quantumcode withBooleanbasisstate isinteresting,itis natural
to question what kind of properties of the Boolean functions used in Lemma 3
should satisfy.