Here is a list of codes related to orthogonal arrays.
Code Description Relation
Binary code Encodes \(K\) states (codewords) in \(n\) binary coordinates and has distance \(d\). Usually denoted as \((n,K,d)\). The distance is the minimum Hamming distance between a pair of distinct codewords. An \((n,K)\) binary code with dual distance \(d^{\perp}\) is an OA\(_{K/2^{d^{\perp}-1}}(d^{\perp}-1,n,2)\) [1][2; Ch. 5].
Denniston code Projective code that is part of a family of \([2^{a+i}+2^i-2^a,3,2^{a+i}-2^a]_{GF(2^a)}\) codes for \(i < a\) constructed using Denniston arcs.
Extended GRS code A GRS code with an additional parity-check coordinate with corresponding evaluation point of zero. In other words, an \([n+1,k,n-k+2]_q\) GRS code whose polynomials are evaluated at the points \((\alpha_1,\cdots,\alpha_n,0)\). The case when \(n=q-1\), multipliers \(v_i=1\), and \(\alpha_i\) are \(i-1\)st powers of a primitive \(n\)th root of unity is an extended narrow-sense RS code.
Generalized RS (GRS) code An \([n,k,n-k+1]_q\) linear code that is a modification of the RS code where codeword polynomials are multiplied by additional prefactors.
Glynn code The unique trace-Hermitian self-dual \([10,5,6]_9\) code, constructed using a 10-arc in \(PG(4,9)\) that is not a rational curve.
Golay code A \([23, 12, 7]\) perfect binary linear code with connections to various areas of mathematics, e.g., lattices [3] and sporadic simple groups [2]. Adding a parity bit to the code results in the self-dual \([24, 12, 8]\) extended Golay code. Up to equivalence, both codes are unique for their respective parameters [4]. Shortening the Golay code yields the \([22,10,8]\), \([22,11,7]\), and \([22,12,6]\) shortened Golay codes [5]. The dual of the Golay code is its \([23,11,8]\) even-weight subcode [6,7]. The extended Golay code is an orthogonal array of strength 7 [8; Exam. 1]
Griesmer code A type of \(q\)-ary code whose parameters satisfy the Griesmer bound with equality.
Hexacode The \([6,3,4]_4\) self-dual MDS code that has connections to projective geometry, lattices [3], and conformal field theory [9]. Puncturing the code yields the perfect \([5,3,3]_4\) quaternary Hamming code known as the shortened hexacode or shorter hexacode [10]. Both codes are sometimes refereed to as Golay codes over \(GF(4)\).
Hirschfeld code A projective geometry code that is an example of an MDS code that is not an RS code; see [11; Exam. 7.6] for the description.
Maximum distance separable (MDS) code A type of \(q\)-ary code whose parameters satisfy the Singleton bound with equality. An MDS code is an OA\(_{1}(k,n,q)\) [12; Thm. 3.3.19].
Mixed code Encodes \(K\) states (codewords) in a string of two or more coordinates, each of which takes values in one of two or more possible groups. Orthogonal arrays generalized to mixed alphabets are called mixed-level orthogonal arrays [13,14], (see [15; Ch. 9]). See Ref. [16] for bounds on mixed orthogonal arrays.
Narrow-sense RS code An \([q-1,k,n-k+1]_q\) RS code whose points \(\alpha_i\) are all \((i-1)\)st powers of a primitive element \(\alpha\) of \(GF(q)\).
Orthogonal array (OA) An orthogonal array, or OA\(_{\lambda}(t,n,q)\), of strength \(t\) with \(q\) levels and \(n\) constraints is a set of \(q\)-ary strings such that any subset of \(t\) coordinates contains every length-\(t\) string an equal number of times \(\lambda\), which is the index of the array.
Ovoid code Member of a \([q^2+1,4,q^2-q]_q\) projective code family that is universally optimal and that is constructed using ovoids in projective space. See [17; pg. 107][18; pg. 192] for further details.
Perfect binary code An \((n,K,2t+1)\) binary code is perfect if parameters \(n\), \(K\), and \(t\) are such that the binary Hamming (a.k.a. sphere-packing) bound \begin{align} \sum_{j=0}^{t} {n \choose j} \leq 2^{n}/K \tag*{(1)}\end{align} becomes an equality. For example, for a code with one logical bit (\(K=2\)) and \(t=1\), the bound becomes \(n+1 \leq 2^{n-1}\). Perfect codes are those for which balls of Hamming radius \(t\) exactly fill the space of all \(n\) binary strings. Perfect distance-three binary codes of length \(n =2^m-1\) are equivalent to binary orthogonal arrays of strength \(t = 2^{m-1}-1\) [19–21].
Perfect-tensor code Block quantum code encoding one subsystem into \(n\) subsystems whose encoding isometry is a perfect tensor. This code stems from an AME\((n,q)\) AME state, or equivalently, a \(((n+1,1,\lfloor (n+1)/2 \rfloor + 1))_{\mathbb{Z}_q}\) code. Orthogonal arrays and \(d\)-uniform quantum states are related [22,23].
Reed-Muller (RM) code Member of the RM\((r,m)\) family of linear binary codes derived from multivariate polynomials. The code parameters are \([2^m,\sum_{j=0}^{r} {m \choose j},2^{m-r}]\), where \(r\) is the order of the code satisfying \(0\leq r\leq m\). First-order RM codes are also called biorthogonal codes, while \(m\)th order RM codes are also called universe codes. Punctured RM codes RM\(^*(r,m)\) are obtained from RM codes by deleting one coordinate from each codeword. RM codes are related to orthogonal arrays [24; Exam. 10.57].
Reed-Solomon (RS) code An \([n,k,n-k+1]_q\) linear code based on polynomials over \(GF(q)\).
Roth-Lempel code Member of a \(q\)-ary linear code family that includes many examples of MDS codes that are not GRS codes.
Semakov-Zinoviev-Zaitsev (SZZ) equidistant code Member of a family that is related to affine resolvable block designs and that is universally optimal.
Single parity-check (SPC) code An \([n,n-1,2]\) linear binary code whose codewords consist of the message string appended with a parity-check bit or parity bit such that the parity (i.e., sum over all coordinates of each codeword) is zero. If the Hamming weight of a message is odd (even), then the parity bit is one (zero). This code requires only one extra bit of overhead and is therefore inexpensive. Its codewords are all even-weight binary strings. Its automorphism group is \(S_n\).
Ternary Golay code A \([11,6,5]_3\) perfect ternary linear code with connections to various areas of mathematics, e.g., lattices [3] and sporadic simple groups [2]. Adding a parity bit to the code results in the self-dual \([12,6,6]_3\) extended ternary Golay code. Up to equivalence, both codes are unique for their respective parameters [4]. The dual of the ternary Golay code is a \([11,5,6]_3\) projective two-weight subcode.
Tetracode The \([4,2,3]_3\) self-dual MDS code that has connections to lattices [3].
Vasilyev code Member of an infinite \((2^{m+1}-1,2^{2n-m},3)\) family of perfect nonlinear codes for any \(m \geq 3\). Constructed by applying a modification of the \((u|u+v)\) construction to a perfect \((2^m-1,2^{n-m},3)\) code, not necessarily linear [2; pg. 77].
\(ED_m\) code Member of the family of \( (c\frac{qt-1}{(t-1,q-1)},qt,ct\frac{q-1}{(t-1,q-1)}) \) \(q\)-ary codes, where \(c,t\geq 1\) and \((a,b)\) is the greatest common divisor of \(a\) and \(b\). Such codes are universally optimal and are related to resolvable block designs.
\([2^m-1,m,2^{m-1}]\) simplex code A member of the code family that is dual to the \([2^m,2^m-m-1,3]\) Hamming family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(m-1,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((2^m,2^m+1)\) simplex spherical code under the antipodal mapping.
\([2^r-1,2^r-r-1,3]\) Hamming code Member of an infinite family of perfect linear codes with parameters \([2^r-1,2^r-r-1, 3]\) for \(r \geq 2\). Their \(r \times (2^r-1) \) parity-check matrix \(H\) has all possible non-zero \(r\)-bit strings as its columns. Adding a parity check yields the \([2^r,2^r-r-1, 4]\) extended Hamming code.
\([7,3,4]\) simplex code Second-smallest member of the simplex code family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(2,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((8,9)\) simplex spherical code under the antipodal mapping.
\([7,4,3]\) Hamming code Second-smallest member of the Hamming code family.
\(q\)-ary parity-check code An \([n,n-1,2]_q\) linear \(q\)-ary code whose codewords consist of the message string appended with a parity-check or zero-sum check digit such that the sum over all coordinates of each codeword is zero.
\(q\)-ary repetition code An \([n,1,n]_q\) code encoding consisting of codewords \((j,j,\cdots,j)\) for \(j \in GF(q)\). The length \(n\) needs to be an odd number, since the receiver will pick the majority to recover the information.
\(q\)-ary sharp configuration A \(q\)-ary code that admits \(m\) different distances between distinct codewords and forms a design of strength \(2m-1\) or greater.
\(q\)-ary simplex code An \([n,m,q^{m-1}]_q\) projective code with \(n=\frac{q^m-1}{q-1}\), denoted as \(S(q,m)\). The columns of the generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(m-1,q)\), with each column being a chosen representative of the corresponding element.

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