Some aspects of codes over rings Peter J. Cameron - PDF document

Some aspects of codes over rings Peter J. Cameron p.j.cameron@qmul.ac.uk Galway, July 2009 This is work by two of my students, Josephine Kusuma and Fatma Al-Kharoosi Summary This was proved by Delsarte for codes over fields. The generalisation is not completely • Codes over rings and orthogonal arrays straightforward. It depends on the following property of rings (which, here, mean finite com- • Z 4 codes and Gray map images mutative rings with identity). • Z 4 codes determined by two binary codes A theorem about rings • Generalisation to Z p n Proposition 2. If I is a proper ideal of the ring R, then the annihilator of R is non-zero. Codes over rings This is false without the assumptions on R , of Rings will always be finite commutative rings with course. It is proved by reducing to the case of local identity . rings, and using the fact that such a ring is equal A (linear) code of length n over R is a submod- to its completion. ule of the free R -module R n . Now the theorem is the case n = 1 of the coding We define the (Hamming) metric d H , the inner result: a code of length 1 is just an ideal of R and product of words, and the dual of a code, over a the dual code is its annihilator. The general case is ring R just as for codes over fields. then proved by a careful induction. It is not true that | Ann ( I ) | = | R | / | I | for any ideal I , and hence not true that | C ⊥ | = | R | n / | C | Orthogonal arrays for any code over the ring R . However this does A code C over an alphabet R is an orthogonal ar- hold for rings such as the integers mod q for posi- ray of strength t if, given any set of t coordinates tive integers q , or for finite fields. i 1 , . . . , i t , and any entries r 1 , . . . , r t ∈ R , there is a constant number of codewords c ∈ C such that The Gray map c i k = r k for k = 1, . . . , t . The Lee metric d L on Z n 4 is defined coordinate- The strength of a code C is the largest t for which wise: C is an orthogonal array of strength t . n ∑ d L ( v , w ) = d L ( v i , w i ) , i = 1 A theorem where the Lee metric on Z 4 is given by the rule that d L ( a , b ) is the number of steps from a to b Theorem 1. The strength of the linear code C over R is one less than the Hamming weight of the dual code when the elements of Z 4 are arranged round a cir- C ⊥ . cle. 1

The Gray map γ is a non-linear map from Z n 4 to Algebraically, there is a homomorphism from C Z 2 n to C 1 with kernel (isomorphic to) C 2 ; so C is an 2 , which is an isometry from the Lee metric on Z n 4 to Z 2 n extension of C 2 by C 1 . 2 . It is also defined coordinatewise: on Z 4 So you should expect cohomology to come in we have somewhere . . . γ ( 0 ) = 00, γ ( 1 ) = 01, γ ( 2 ) = 11, γ ( 3 ) = 10. It was introduced by Hammons et al. in their clas- The class C ( C 1 , C 2 ) sic paper showing that certain nonlinear binary We note that C 1 ≤ C 2 . For, given any word c ∈ C 1 , let c ′ be a word in C mapping onto c ; then 2 c ′ codes such as the Nordstrom–Robinson, Preparata and Kerdock codes are Gray map images of linear has all entries 0 or 2 and produces the word c ∈ C 2 . Z 4 -codes. Given binary codes C 1 ≤ C 2 , let C ( C 1 , C 2 ) be the set of all Z 4 -codes C corresponding as above to the pair C 1 , C 2 . The Gray map Proposition 5. If the length is n, and dim ( C i ) = k i 2 11 ✉ ✉ for i = 1, 2 , then |C ( C 1 , C 2 ) | = 2 k 1 ( n − k 2 ) . � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ Given C 1 and C 2 , what can we say about prop- Z 2 3 Z 4 1 10 01 ✉ ✉ ✉ ✉ ❅ � ❅ � 2 erties of the codes in C ( C 1 , C 2 ) ? ❅ � ❅ � ❅ � ❅ � ✉ ✉ 0 00 Generator matrices The code C has a generator matrix of the form � � I X Y A theorem and a conjecture . O 2 I 2 Z Conjecture 3. Let C be a linear code over Z 4 and C ′ The generator matrices of C 1 and C 2 are respec- its Gray map image. Then the strength of C ′ is one less � � I X Y � � tively I X Y and (where the en- than the minimum Lee weight of C ⊥ . O I Z tries are read mod 2). Note that the strength of C is one less than the We can assume that X is a zero-one matrix. Then minimum Hamming weight of C ⊥ . Moreover, if C and C ′ have strength t and t ′ re- Y is only determined mod 2 by C 1 and C 2 , so the spectively, then it is known that t ≤ t ′ ≤ 2 t + 1. codes in C ( C 1 , C 2 ) are found by adding 0 or 2 to the elements of Y . (This would follow from the truth of the conjec- Since Y is k 1 × ( n − k 2 ) , where k i = dim ( C i ) , this ture.) gives the formula for |C ( C 1 , C 2 ) | . Theorem 4. Let C be a linear code over Z 4 and C ′ its Gray map image. Then the strength of C ′ is at most the minimum Lee weight of C ⊥ minus one. Weight enumerators The symmetrized weight enumerator of a Z 4 -code C is the three-variable homogeneous polynomial A classification of Z 4 -codes With any Z 4 -code C , we can associate a pair x n 0 ( c ) y n 2 ( c ) z n 1 ( c )+ n 3 ( c ) . ∑ ( C 1 , C 2 ) of binary codes as follows. (This is a spe- c ∈ C cial case of a construction due to Eric Lander). Apart from renormalisation, we obtain the weight • C 1 is obtained by reading the entries in words enumerators of C 1 and C 2 by the substitutions x → of C mod 2, so that 0 and 2 become 0, 1 and 3 x , y → x , z → y and x → x , y → y and z → 0 become 1. respectively. • C 2 is obtained by considering just those words The Lee weight enumerator of C , and hence the of C with entries 0 and 2 only, and replacing 0 weight enumerator of the Gray map image, is ob- tained by the substitution x → x 2 , y → y 2 , z → xy . by 0 and 2 by 1. 2

Theorem 6. The average of the symmetrized weight The derivations modulo inner derivations form a group, the first cohomology group H 1 ( G , A ) , enumerators of the codes in C ( C 1 , C 2 ) is whose elements correspond bijectively to the con- W C 1 ( x + y , 2 z ) − ( x + y ) n � + W C 2 ( x , y ) . | C 2 | jugacy classes of complements of the normal sub- � 2 n group A in the semidirect product A : G . If A is a vector space and G a linear group, then Weight enumerators, continued A : G is a group of affine transformations of A ; the Carrie Rutherford and I are currently trying to stabilizer of the zero vector is a complement, and a obtain further global information about this; in complement is conjugate to G if and only if it fixes particular, the “variance” of the weight enumer- a vector. ators of codes in C ( C 1 , C 2 ) . A case study Fatma Al-Kharoosi examined this situation lo- A very interesting case is that in which C 1 = cally, and showed that there are only a limited C 2 is the extended Hamming code of length 8. number of possibilities for the way that the s.w.e. The class C ( C 1 , C 2 ) includes the “octacode” whose changes in moving from one code in the class to a Gray map image is the non-linear Nordstrom– neighbouring one. Robinson code of length 16. A detailed example is given later. The class C in this case admits the group G = ( Z 7 2 ) : AGL ( 3, 2 ) (the first factor corresponds to C ( C 1 , C 2 ) as an affine space coordinate sign changes, the second is the com- The fact that |C ( C 1 , C 2 ) | is a power of 2 is not a mon automorphism group of C 1 and C 2 ). coincidence: the group C ∗ 1 ⊗ ( Z n 2 / C 2 ) acts on this The cohomology group H 1 ( G , W ) is non-zero, set by translation. ( C ∗ 1 is the dual space of C 1 .) and indeed the class C realises an outer deriva- For C ∗ 1 ⊗ Z n 2 acts on C by the rule tion. ( f ⊗ w )( c ) = c + d ( f ( c mod 2 )) w A case study, continued where d is the “doubling” map 0 → 0, 1 → 2 from The table gives the orbit lengths of G on C , Z 2 to Z 4 , and the kernel of the action is C ∗ 1 ⊗ C 2 . the symmetrized weight enumerator of a code in So if we fix a reference code in C to act as origin, each orbit, and the number of orbits of the sub- there is a bijection between C and C ∗ 1 ⊗ ( Z n 2 / C 2 ) . group AGL ( 3, 2 ) (the automorphism group of the extended Hamming code). Here Another group action F ( x , y , z ) = x 8 + 14 x 4 y 4 + y 8 + 16 z 8 + 112 xyz 4 ( x 2 + y 2 ) It is clear that C is invariant under Aut ( C 1 ) ∩ Aut ( C 2 ) , the common automorphisms of C 1 and is the weight enumerator of the octacode, and C 2 . Also, 3 is a unit in Z 4 , so multiplying any set E ( x , y , z ) = 4 z 4 ( x − y ) 4 . of coordinate by 3 maps each code in C to another with the same symmetrized weight enumerator. The data Multiplying all coordinates by 3 fixes all the codes, so the group Z n − 1 acts. 2 Orbit SWE #perm orbits These two groups generate their semidirect 7168 F+5E 19 product ( Z n − 1 ) : ( Aut ( C 1 ) ∩ Aut ( C 2 )) . 896 F+6E 7 2 21504 F+4E 24 First cohomology 21504 F+3E 27 Let A be an abelian group, and G a group acting 3584 F+4E 14 on A . 896 F+4E 4 A derivation is a map d : G → A satisfying 7168 F+2E 8 d ( g 1 g 2 ) = d ( g 1 ) g 2 + d ( g 2 ) . It is inner if there is 2688 F+2E 8 an element a ∈ A such that d ( g ) = a g − a . 128 F 3 3